9.4: Solve Quadratic Equations Using the Quadratic Formula (2024)

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    Learning Objectives

    By the end of this section, you will be able to:

    • Solve quadratic equations using the Quadratic Formula
    • Use the discriminant to predict the number and type of solutions of a quadratic equation
    • Identify the most appropriate method to use to solve a quadratic equation
    Be Prepared 9.7

    Before you get started, take this readiness quiz.

    Evaluate b24abb24ab when a=3a=3 and b=−2.b=−2.
    If you missed this problem, review Example 1.21.

    Be Prepared 9.8

    Simplify: 108.108.
    If you missed this problem, review Example 8.13.

    Be Prepared 9.9

    Simplify: 50.50.
    If you missed this problem, review Example 8.76.

    Solve Quadratic Equations Using the Quadratic Formula

    When we solved quadratic equations in the last section by completing the square, we took the same steps every time. By the end of the exercise set, you may have been wondering ‘isn’t there an easier way to do this?’ The answer is ‘yes’. Mathematicians look for patterns when they do things over and over in order to make their work easier. In this section we will derive and use a formula to find the solution of a quadratic equation.

    We have already seen how to solve a formula for a specific variable ‘in general’, so that we would do the algebraic steps only once, and then use the new formula to find the value of the specific variable. Now we will go through the steps of completing the square using the general form of a quadratic equation to solve a quadratic equation for x.

    We start with the standard form of a quadratic equation and solve it for x by completing the square.

    9.4: Solve Quadratic Equations Using the Quadratic Formula (2)
    Isolate the variable terms on one side. 9.4: Solve Quadratic Equations Using the Quadratic Formula (3)
    Make the coefficient of x2x2 equal to 1, by
    dividing by a.
    9.4: Solve Quadratic Equations Using the Quadratic Formula (4)
    Simplify. 9.4: Solve Quadratic Equations Using the Quadratic Formula (5)
    To complete the square, find (12·ba)2(12·ba)2 and add it to both sides of the equation.
    (12ba)2=b24a2(12ba)2=b24a2 9.4: Solve Quadratic Equations Using the Quadratic Formula (6)
    The left side is a perfect square, factor it. 9.4: Solve Quadratic Equations Using the Quadratic Formula (7)
    Find the common denominator of the right
    side and write equivalent fractions with
    the common denominator.
    9.4: Solve Quadratic Equations Using the Quadratic Formula (8)
    Simplify. 9.4: Solve Quadratic Equations Using the Quadratic Formula (9)
    Combine to one fraction. 9.4: Solve Quadratic Equations Using the Quadratic Formula (10)
    Use the square root property. 9.4: Solve Quadratic Equations Using the Quadratic Formula (11)
    Simplify the radical. 9.4: Solve Quadratic Equations Using the Quadratic Formula (12)
    Add b2ab2a to both sides of the equation. 9.4: Solve Quadratic Equations Using the Quadratic Formula (13)
    Combine the terms on the right side. 9.4: Solve Quadratic Equations Using the Quadratic Formula (14)
    This equation is the Quadratic Formula.
    Quadratic Formula

    The solutions to a quadratic equation of the form ax2 + bx + c = 0, where a0a0 are given by the formula:

    x=b±b24ac2ax=b±b24ac2a

    To use the Quadratic Formula, we substitute the values of a, b, and c from the standard form into the expression on the right side of the formula. Then we simplify the expression. The result is the pair of solutions to the quadratic equation.

    Notice the formula is an equation. Make sure you use both sides of the equation.

    Example 9.21

    How to Solve a Quadratic Equation Using the Quadratic Formula

    Solve by using the Quadratic Formula: 2x2+9x5=0.2x2+9x5=0.

    Answer

    9.4: Solve Quadratic Equations Using the Quadratic Formula (15) 9.4: Solve Quadratic Equations Using the Quadratic Formula (16) 9.4: Solve Quadratic Equations Using the Quadratic Formula (17) 9.4: Solve Quadratic Equations Using the Quadratic Formula (18)

    Try It 9.41

    Solve by using the Quadratic Formula: 3y25y+2=03y25y+2=0.

    Try It 9.42

    Solve by using the Quadratic Formula: 4z2+2z6=04z2+2z6=0.

    How To

    Solve a quadratic equation using the quadratic formula.

    1. Step 1. Write the quadratic equation in standard form, ax2 + bx + c = 0. Identify the values of a, b, and c.
    2. Step 2. Write the Quadratic Formula. Then substitute in the values of a, b, and c.
    3. Step 3. Simplify.
    4. Step 4. Check the solutions.

    If you say the formula as you write it in each problem, you’ll have it memorized in no time! And remember, the Quadratic Formula is an EQUATION. Be sure you start with “x =”.

    Example 9.22

    Solve by using the Quadratic Formula: x26x=−5.x26x=−5.

    Answer
    9.4: Solve Quadratic Equations Using the Quadratic Formula (19)
    Write the equation in standard form by adding
    5 to each side.
    9.4: Solve Quadratic Equations Using the Quadratic Formula (20)
    This equation is now in standard form. 9.4: Solve Quadratic Equations Using the Quadratic Formula (21)
    Identify the values of a,b,c.a,b,c. 9.4: Solve Quadratic Equations Using the Quadratic Formula (22)
    Write the Quadratic Formula. 9.4: Solve Quadratic Equations Using the Quadratic Formula (23)
    Then substitute in the values of a,b,c.a,b,c. 9.4: Solve Quadratic Equations Using the Quadratic Formula (24)
    Simplify. 9.4: Solve Quadratic Equations Using the Quadratic Formula (25)
    9.4: Solve Quadratic Equations Using the Quadratic Formula (26)
    9.4: Solve Quadratic Equations Using the Quadratic Formula (27)
    Rewrite to show two solutions. 9.4: Solve Quadratic Equations Using the Quadratic Formula (28)
    Simplify. 9.4: Solve Quadratic Equations Using the Quadratic Formula (29)
    9.4: Solve Quadratic Equations Using the Quadratic Formula (30)
    Check:

    9.4: Solve Quadratic Equations Using the Quadratic Formula (31)

    Try It 9.43

    Solve by using the Quadratic Formula: a22a=15a22a=15.

    Try It 9.44

    Solve by using the Quadratic Formula: b2+24=−10bb2+24=−10b.

    When we solved quadratic equations by using the Square Root Property, we sometimes got answers that had radicals. That can happen, too, when using the Quadratic Formula. If we get a radical as a solution, the final answer must have the radical in its simplified form.

    Example 9.23

    Solve by using the Quadratic Formula: 2x2+10x+11=0.2x2+10x+11=0.

    Answer
    9.4: Solve Quadratic Equations Using the Quadratic Formula (32)
    This equation is in standard form. 9.4: Solve Quadratic Equations Using the Quadratic Formula (33)
    Identify the values of a, b, and c. 9.4: Solve Quadratic Equations Using the Quadratic Formula (34)
    Write the Quadratic Formula. 9.4: Solve Quadratic Equations Using the Quadratic Formula (35)
    Then substitute in the values of a, b, and c. 9.4: Solve Quadratic Equations Using the Quadratic Formula (36)
    Simplify. 9.4: Solve Quadratic Equations Using the Quadratic Formula (37)
    9.4: Solve Quadratic Equations Using the Quadratic Formula (38)
    Simplify the radical. 9.4: Solve Quadratic Equations Using the Quadratic Formula (39)
    Factor out the common factor in the numerator. 9.4: Solve Quadratic Equations Using the Quadratic Formula (40)
    Remove the common factors. 9.4: Solve Quadratic Equations Using the Quadratic Formula (41)
    Rewrite to show two solutions. 9.4: Solve Quadratic Equations Using the Quadratic Formula (42)
    Check:
    We leave the check for you!
    Try It 9.45

    Solve by using the Quadratic Formula: 3m2+12m+7=03m2+12m+7=0.

    Try It 9.46

    Solve by using the Quadratic Formula: 5n2+4n4=05n2+4n4=0.

    When we substitute a, b, and c into the Quadratic Formula and the radicand is negative, the quadratic equation will have imaginary or complex solutions. We will see this in the next example.

    Example 9.24

    Solve by using the Quadratic Formula: 3p2+2p+9=0.3p2+2p+9=0.

    Answer
    9.4: Solve Quadratic Equations Using the Quadratic Formula (43)
    This equation is in standard form 9.4: Solve Quadratic Equations Using the Quadratic Formula (44)
    Identify the values of a,b,c.a,b,c. 9.4: Solve Quadratic Equations Using the Quadratic Formula (45)
    Write the Quadratic Formula. 9.4: Solve Quadratic Equations Using the Quadratic Formula (46)
    Then substitute in the values of a,b,ca,b,c. 9.4: Solve Quadratic Equations Using the Quadratic Formula (47)
    Simplify. 9.4: Solve Quadratic Equations Using the Quadratic Formula (48)
    9.4: Solve Quadratic Equations Using the Quadratic Formula (49)
    Simplify the radical using complex numbers. 9.4: Solve Quadratic Equations Using the Quadratic Formula (50)
    Simplify the radical. 9.4: Solve Quadratic Equations Using the Quadratic Formula (51)
    Factor the common factor in the numerator. 9.4: Solve Quadratic Equations Using the Quadratic Formula (52)
    Remove the common factors. 9.4: Solve Quadratic Equations Using the Quadratic Formula (53)
    Rewrite in standard a+bia+bi form. 9.4: Solve Quadratic Equations Using the Quadratic Formula (54)
    Write as two solutions. 9.4: Solve Quadratic Equations Using the Quadratic Formula (55)
    Try It 9.48

    Solve by using the Quadratic Formula: 5b2+2b+4=05b2+2b+4=0.

    Remember, to use the Quadratic Formula, the equation must be written in standard form, ax2 + bx + c = 0. Sometimes, we will need to do some algebra to get the equation into standard form before we can use the Quadratic Formula.

    Example 9.25

    Solve by using the Quadratic Formula: x(x+6)+4=0.x(x+6)+4=0.

    Answer

    Our first step is to get the equation in standard form.

    9.4: Solve Quadratic Equations Using the Quadratic Formula (56)
    Distribute to get the equation in standard form. 9.4: Solve Quadratic Equations Using the Quadratic Formula (57)
    This equation is now in standard form 9.4: Solve Quadratic Equations Using the Quadratic Formula (58)
    Identify the values of a,b,c.a,b,c. 9.4: Solve Quadratic Equations Using the Quadratic Formula (59)
    Write the Quadratic Formula. 9.4: Solve Quadratic Equations Using the Quadratic Formula (60)
    Then substitute in the values of a,b,ca,b,c. 9.4: Solve Quadratic Equations Using the Quadratic Formula (61)
    Simplify. 9.4: Solve Quadratic Equations Using the Quadratic Formula (62)
    9.4: Solve Quadratic Equations Using the Quadratic Formula (63)
    Simplify the radical. 9.4: Solve Quadratic Equations Using the Quadratic Formula (64)
    Factor the common factor in the numerator. 9.4: Solve Quadratic Equations Using the Quadratic Formula (65)
    Remove the common factors. 9.4: Solve Quadratic Equations Using the Quadratic Formula (66)
    Write as two solutions. 9.4: Solve Quadratic Equations Using the Quadratic Formula (67)
    Check:
    We leave the check for you!
    Try It 9.49

    Solve by using the Quadratic Formula: x(x+2)5=0.x(x+2)5=0.

    Try It 9.50

    Solve by using the Quadratic Formula: 3y(y2)3=0.3y(y2)3=0.

    When we solved linear equations, if an equation had too many fractions we cleared the fractions by multiplying both sides of the equation by the LCD. This gave us an equivalent equation—without fractions— to solve. We can use the same strategy with quadratic equations.

    Example 9.26

    Solve by using the Quadratic Formula: 12u2+23u=13.12u2+23u=13.

    Answer

    Our first step is to clear the fractions.

    9.4: Solve Quadratic Equations Using the Quadratic Formula (68)
    Multiply both sides by the LCD, 6, to clear the fractions. 9.4: Solve Quadratic Equations Using the Quadratic Formula (69)
    Multiply. 9.4: Solve Quadratic Equations Using the Quadratic Formula (70)
    Subtract 2 to get the equation in standard form. 9.4: Solve Quadratic Equations Using the Quadratic Formula (71)
    Identify the values of a, b, and c. 9.4: Solve Quadratic Equations Using the Quadratic Formula (72)
    Write the Quadratic Formula. 9.4: Solve Quadratic Equations Using the Quadratic Formula (73)
    Then substitute in the values of a, b, and c. 9.4: Solve Quadratic Equations Using the Quadratic Formula (74)
    Simplify. 9.4: Solve Quadratic Equations Using the Quadratic Formula (75)
    9.4: Solve Quadratic Equations Using the Quadratic Formula (76)
    Simplify the radical. 9.4: Solve Quadratic Equations Using the Quadratic Formula (77)
    Factor the common factor in the numerator. 9.4: Solve Quadratic Equations Using the Quadratic Formula (78)
    Remove the common factors. 9.4: Solve Quadratic Equations Using the Quadratic Formula (79)
    Rewrite to show two solutions. 9.4: Solve Quadratic Equations Using the Quadratic Formula (80)
    Check:
    We leave the check for you!
    Try It 9.51

    Solve by using the Quadratic Formula: 14c213c=11214c213c=112.

    Try It 9.52

    Solve by using the Quadratic Formula: 19d212d=1319d212d=13.

    Think about the equation (x − 3)2 = 0. We know from the Zero Product Property that this equation has only one solution,
    x = 3.

    We will see in the next example how using the Quadratic Formula to solve an equation whose standard form is a perfect square trinomial equal to 0 gives just one solution. Notice that once the radicand is simplified it becomes 0 , which leads to only one solution.

    Example 9.27

    Solve by using the Quadratic Formula: 4x220x=−25.4x220x=−25.

    Answer
    9.4: Solve Quadratic Equations Using the Quadratic Formula (81)
    Add 25 to get the equation in standard form. 9.4: Solve Quadratic Equations Using the Quadratic Formula (82)
    Identify the values of a, b, and c. 9.4: Solve Quadratic Equations Using the Quadratic Formula (83)
    Write the quadratic formula. 9.4: Solve Quadratic Equations Using the Quadratic Formula (84)
    Then substitute in the values of a, b, and c. 9.4: Solve Quadratic Equations Using the Quadratic Formula (85)
    Simplify. 9.4: Solve Quadratic Equations Using the Quadratic Formula (86)
    9.4: Solve Quadratic Equations Using the Quadratic Formula (87)
    Simplify the radical. 9.4: Solve Quadratic Equations Using the Quadratic Formula (88)
    Simplify the fraction. 9.4: Solve Quadratic Equations Using the Quadratic Formula (89)
    Check:
    We leave the check for you!

    Did you recognize that 4x2 − 20x + 25 is a perfect square trinomial. It is equivalent to (2x − 5)2? If you solve
    4x2 − 20x + 25 = 0 by factoring and then using the Square Root Property, do you get the same result?

    Try It 9.53

    Solve by using the Quadratic Formula: r2+10r+25=0.r2+10r+25=0.

    Try It 9.54

    Solve by using the Quadratic Formula: 25t240t=−16.25t240t=−16.

    Use the Discriminant to Predict the Number and Type of Solutions of a Quadratic Equation

    When we solved the quadratic equations in the previous examples, sometimes we got two real solutions, one real solution, and sometimes two complex solutions. Is there a way to predict the number and type of solutions to a quadratic equation without actually solving the equation?

    Yes, the expression under the radical of the Quadratic Formula makes it easy for us to determine the number and type of solutions. This expression is called the discriminant.

    Discriminant

    9.4: Solve Quadratic Equations Using the Quadratic Formula (90)

    Let’s look at the discriminant of the equations in some of the examples and the number and type of solutions to those quadratic equations.

    Quadratic Equation
    (in standard form)
    Discriminant
    b24acb24ac
    Value of the Discriminant Number and Type of solutions
    2x2+9x5=02x2+9x5=0 924·2(−5)121924·2(−5)121 + 2 real
    4x220x+25=04x220x+25=0 (−20)24·4·250(−20)24·4·250 0 1 real
    3p2+2p+9=03p2+2p+9=0 224·3·9104224·3·9104 2 complex

    9.4: Solve Quadratic Equations Using the Quadratic Formula (91)

    Using the Discriminant, b2 − 4ac, to Determine the Number and Type of Solutions of a Quadratic Equation

    For a quadratic equation of the form ax2 + bx + c = 0, a0,a0,

    • If b2 − 4ac > 0, the equation has 2 real solutions.
    • if b2 − 4ac = 0, the equation has 1 real solution.
    • if b2 − 4ac < 0, the equation has 2 complex solutions.
    Example 9.28

    Determine the number of solutions to each quadratic equation.

    3x2+7x9=03x2+7x9=0 5n2+n+4=05n2+n+4=0 9y26y+1=0.9y26y+1=0.

    Answer

    To determine the number of solutions of each quadratic equation, we will look at its discriminant.

    3x2+7x9=03x2+7x9=0
    The equation is in standard form, identify a, b, and c. a=3,b=7,c=−9a=3,b=7,c=−9
    Write the discriminant. b24acb24ac
    Substitute in the values of a, b, and c. (7)24·3·(−9)(7)24·3·(−9)
    Simplify. 49+10849+108
    157157

    Since the discriminant is positive, there are 2 real solutions to the equation.

    5n2+n+4=05n2+n+4=0
    The equation is in standard form, identify a, b, and c. a=5,b=1,c=4a=5,b=1,c=4
    Write the discriminant. b24acb24ac
    Substitute in the values of a, b, and c. (1)24·5·4(1)24·5·4
    Simplify. 180180
    −79−79

    Since the discriminant is negative, there are 2 complex solutions to the equation.

    9y26y+1=09y26y+1=0
    The equation is in standard form, identify a, b, and c. a=9,b=−6,c=1a=9,b=−6,c=1
    Write the discriminant. b24acb24ac
    Substitute in the values of a, b, and c. (−6)24·9·1(−6)24·9·1
    Simplify. 36363636
    00

    Since the discriminant is 0, there is 1 real solution to the equation.

    Try It 9.55

    Determine the numberand type of solutions to each quadratic equation.

    8m23m+6=08m23m+6=0 5z2+6z2=05z2+6z2=0 9w2+24w+16=0.9w2+24w+16=0.

    Try It 9.56

    Determine the number and type of solutions to each quadratic equation.

    b2+7b13=0b2+7b13=0 5a26a+10=05a26a+10=0 4r220r+25=0.4r220r+25=0.

    Identify the Most Appropriate Method to Use to Solve a Quadratic Equation

    We summarize the four methods that we have used to solve quadratic equations below.

    Methods for Solving Quadratic Equations
    1. Factoring
    2. Square Root Property
    3. Completing the Square
    4. Quadratic Formula

    Given that we have four methods to use to solve a quadratic equation, how do you decide which one to use? Factoring is often the quickest method and so we try it first. If the equation is ax2=kax2=k or a(xh)2=ka(xh)2=k we use the Square Root Property. For any other equation, it is probably best to use the Quadratic Formula. Remember, you can solve any quadratic equation by using the Quadratic Formula, but that is not always the easiest method.

    What about the method of Completing the Square? Most people find that method cumbersome and prefer not to use it. We needed to include it in the list of methods because we completed the square in general to derive the Quadratic Formula. You will also use the process of Completing the Square in other areas of algebra.

    How To

    Identify the most appropriate method to solve a quadratic equation.

    1. Step 1. Try Factoring first. If the quadratic factors easily, this method is very quick.
    2. Step 2. Try the Square Root Property next. If the equation fits the form ax2=kax2=k or a(xh)2=k,a(xh)2=k, it can easily be solved by using the Square Root Property.
    3. Step 3. Use the Quadratic Formula. Any other quadratic equation is best solved by using the Quadratic Formula.

    The next example uses this strategy to decide how to solve each quadratic equation.

    Example 9.29

    Identify the most appropriate method to use to solve each quadratic equation.

    5z2=175z2=17 4x212x+9=04x212x+9=0 8u2+6u=11.8u2+6u=11.

    Answer


    5z2=175z2=17

    Since the equation is in the ax2=k,ax2=k, the most appropriate method is to use the Square Root Property.


    4x212x+9=04x212x+9=0

    We recognize that the left side of the equation is a perfect square trinomial, and so factoring will be the most appropriate method.

    8u2+6u=118u2+6u=11
    Put the equation in standard form. 8u2+6u11=08u2+6u11=0

    While our first thought may be to try factoring, thinking about all the possibilities for trial and error method leads us to choose the Quadratic Formula as the most appropriate method.

    Try It 9.57

    Identify the most appropriate method to use to solve each quadratic equation.

    x2+6x+8=0x2+6x+8=0 (n3)2=16(n3)2=16 5p26p=9.5p26p=9.

    Try It 9.58

    Identify the most appropriate method to use to solve each quadratic equation.

    8a2+3a9=08a2+3a9=0 4b2+4b+1=04b2+4b+1=0 5c2=125.5c2=125.

    Media

    Access these online resources for additional instruction and practice with using the Quadratic Formula.

    Section 9.3 Exercises

    Practice Makes Perfect

    Solve Quadratic Equations Using the Quadratic Formula

    In the following exercises, solve by using the Quadratic Formula.

    113.

    4 m 2 + m 3 = 0 4 m 2 + m 3 = 0

    114.

    4 n 2 9 n + 5 = 0 4 n 2 9 n + 5 = 0

    115.

    2 p 2 7 p + 3 = 0 2 p 2 7 p + 3 = 0

    116.

    3 q 2 + 8 q 3 = 0 3 q 2 + 8 q 3 = 0

    117.

    p 2 + 7 p + 12 = 0 p 2 + 7 p + 12 = 0

    118.

    q 2 + 3 q 18 = 0 q 2 + 3 q 18 = 0

    119.

    r 2 8 r = 33 r 2 8 r = 33

    120.

    t 2 + 13 t = −40 t 2 + 13 t = −40

    121.

    3 u 2 + 7 u 2 = 0 3 u 2 + 7 u 2 = 0

    122.

    2 p 2 + 8 p + 5 = 0 2 p 2 + 8 p + 5 = 0

    123.

    2 a 2 6 a + 3 = 0 2 a 2 6 a + 3 = 0

    124.

    5 b 2 + 2 b 4 = 0 5 b 2 + 2 b 4 = 0

    125.

    x 2 + 8 x 4 = 0 x 2 + 8 x 4 = 0

    126.

    y 2 + 4 y 4 = 0 y 2 + 4 y 4 = 0

    127.

    3 y 2 + 5 y 2 = 0 3 y 2 + 5 y 2 = 0

    128.

    6 x 2 + 2 x 20 = 0 6 x 2 + 2 x 20 = 0

    129.

    2 x 2 + 3 x + 3 = 0 2 x 2 + 3 x + 3 = 0

    130.

    2 x 2 x + 1 = 0 2 x 2 x + 1 = 0

    131.

    8 x 2 6 x + 2 = 0 8 x 2 6 x + 2 = 0

    132.

    8 x 2 4 x + 1 = 0 8 x 2 4 x + 1 = 0

    133.

    ( v + 1 ) ( v 5 ) 4 = 0 ( v + 1 ) ( v 5 ) 4 = 0

    134.

    ( x + 1 ) ( x 3 ) = 2 ( x + 1 ) ( x 3 ) = 2

    135.

    ( y + 4 ) ( y 7 ) = 18 ( y + 4 ) ( y 7 ) = 18

    136.

    ( x + 2 ) ( x + 6 ) = 21 ( x + 2 ) ( x + 6 ) = 21

    137.

    1 3 m 2 + 1 12 m = 1 4 1 3 m 2 + 1 12 m = 1 4

    138.

    1 3 n 2 + n = 1 2 1 3 n 2 + n = 1 2

    139.

    3 4 b 2 + 1 2 b = 3 8 3 4 b 2 + 1 2 b = 3 8

    140.

    1 9 c 2 + 2 3 c = 3 1 9 c 2 + 2 3 c = 3

    141.

    16 c 2 + 24 c + 9 = 0 16 c 2 + 24 c + 9 = 0

    142.

    25 d 2 60 d + 36 = 0 25 d 2 60 d + 36 = 0

    143.

    25 q 2 + 30 q + 9 = 0 25 q 2 + 30 q + 9 = 0

    144.

    16 y 2 + 8 y + 1 = 0 16 y 2 + 8 y + 1 = 0

    Use the Discriminant to Predict the Number of Real Solutions of a Quadratic Equation

    In the following exercises, determine the number of real solutions for each quadratic equation.

    145.

    4x25x+16=04x25x+16=0 36y2+36y+9=036y2+36y+9=0 6m2+3m5=06m2+3m5=0

    146.

    9v215v+25=09v215v+25=0 100w2+60w+9=0100w2+60w+9=0 5c2+7c10=05c2+7c10=0

    147.

    r2+12r+36=0r2+12r+36=0 8t211t+5=08t211t+5=0 3v25v1=03v25v1=0

    148.

    25p2+10p+1=025p2+10p+1=0 7q23q6=07q23q6=0 7y2+2y+8=07y2+2y+8=0

    Identify the Most Appropriate Method to Use to Solve a Quadratic Equation

    In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.

    149.


    x25x24=0x25x24=0
    (y+5)2=12(y+5)2=12
    14m2+3m=1114m2+3m=11

    150.


    (8v+3)2=81(8v+3)2=81
    w29w22=0w29w22=0
    4n210n=64n210n=6

    151.


    6a2+14a=206a2+14a=20
    (x14)2=516(x14)2=516
    y22y=8y22y=8

    152.


    8b2+15b=48b2+15b=4
    59v223v=159v223v=1
    (w+43)2=29(w+43)2=29

    Writing Exercises

    153.

    Solve the equation x2+10x=120x2+10x=120

    by completing the square

    using the Quadratic Formula

    Which method do you prefer? Why?

    154.

    Solve the equation 12y2+23y=2412y2+23y=24

    by completing the square

    using the Quadratic Formula

    Which method do you prefer? Why?

    Self Check

    After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    9.4: Solve Quadratic Equations Using the Quadratic Formula (92)

    What does this checklist tell you about your mastery of this section? What steps will you take to improve?

    9.4: Solve Quadratic Equations Using the Quadratic Formula (2024)
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