Intermediate Algebra – 20

HFCC Math Lab

SOLVING QUADRATIC EQUATIONS USING THE QUADRATIC FORMULA

Quadratic equations can be solved by a number of methods: 1. Factoring – often the fastest method, but can’t be used if the equation can’t be factored. 2. Square root method – can only be used if the quadratic equation is written as a perfect square trinomial. 3. Completing the square – works on every quadratic equation, but requires a detailed algebraic process to complete the square. 4. Quadratic formula – works on every quadratic equation and only requires substitution and arithmetic simplification. This handout will illustrate solving quadratic equations using the quadratic formula.

Quadratic formula: The solutions of the quadratic equation ax2 bx c 0 where

a

0 are x

b

b 2 4ac . 2a

Note: The quadratic formula is derived by completing the square on the general quadratic equation in standard from, ax2 bx c 0 . This derivation is shown at the end of the handout for the interested reader.

Method for using the quadratic formula: 1. Write the given quadratic equation in standard form, that is, in the form ax2 bx c 0 . 2. Identify the coefficients a, b, and c in the quadratic equation in standard form. 3. Substitute these coefficients into the quadratic formula and simplify the expression.

Revised 04/10

1

Ex 1: Solve 10 x 2

3 x using the quadratic formula.

10 x 2

3 x

The original equation.

10 x 2

x 3 0

Write the equation in standard form by adding 3 and x to both sides of the equation.

a 10, b 1, and c

x

x

b

b 2 4ac 2a

State the quadratic formula.

1

12 4(10)( 3) 2(10)

Substitute the coefficients in the quadratic formula.

1

x

121 20

Simplify.

x

1 11 20

x

1 11 or x 20

x

Simplify.

1 or x 2

Ex 2: Solve

Identify the coefficients a, b, and c .

3

1 11 20

Separate the

3 5

into two cases.

Simplify.

3 x using the quadratic formula. 1 4x 2

4x 1

3 4x

4x 1

1 1

4x 1

x 2

Multiply both sides by the LCD to clear the denominators.

3 4x

2 x2

Simplify by reducing.

2 x2 4 x 3 0 a

2, b

Revised 04/10

4, and c

Write in standard form. 3

Identify the coefficients a, b, and c .

2

x

x

4

State the quadratic formula.

( 4) 2 4(2)( 3) 2(2)

( 4)

x

x

b 2 4ac 2a

b

40

Simplify.

4 4 2 10 4 2 2

x

Substitute the coefficients in the quadratic formula.

Simplify.

10

Factor the common factor 2 in the numerator.

4 1

2 2 x

10

Reduce the common factor 2.

4 2

x

2

10

The simplified answer.

2

Ex 3: (2 x 3)(2 x 3) (2 x 3)(2 x 3) 4 x2 6 x 6 x 9

14 using the quadratic formula. 14 14

The original equation. Multiply the binomials.

4 x2 0 x 5 0

Simplify and write in standard form.

a

Identify the coefficients a, b, and c .

x

x

4, b 0, and c 5

b

(0)

Revised 04/10

b 2 4ac 2a (0) 2 4(4)(5) 2(4)

State the quadratic formula.

Substitute the coefficients in the quadratic formula.

3

x

80 8

Simplify.

x

( 1)(16)(5) 8

Factor the radicand.

x

4i 5 8

x

4i 5 8

Simplify and use

1 i.

1

Reduce the common factor 4.

2

5 i 2

x

The simplified answer.

Derivation of the quadratic formula: ax 2 bx c

x2

b c x a a

x2

b x a

1 b 2 a

2

0, a

The quadratic equation in standard form.

0

Divide both sides by a.

0 c a

Add

c to both sides of the equation. a

b2 4a 2

The term needed to complete the square is one-half the square of the coefficient of x .

x2

b b2 x a 4a 2

c a

x2

b b2 x a 4a 2

4ac 4a 2

b b2 x a 4a 2

b 2 4ac 4a 2

x

2

Revised 04/10

b2 4a 2 b2 4a 2

Add

b2 to both sides. 4a 2

4a 2 is the LCD for the right side.

Simply the expression on the right side.

4

2

b 2a

x

b2

4ac 4a 2

Factor the perfect square trinomial on the left side.

x

b 2a

b 2a

x

Apply the square root rule and simplify.

b 2 4ac 2a

Add

b 2 4ac 2a

b

x

b 2 4ac 2a

b to both sides. 2a

Simplify the right side by writing both terms over the common denominator 2a .

Exercises: Solve the following equations using the quadratic formula. 1. 3x2 5x 12 0

2. 8x2 2 x 21 0

3. x2 10 x 22

4. x2 2

5.

1 2 x 3

7. x2

1 x 2 x

5 6

11.

3x 2

6. x 2 10 x 3

4x2 2 x

9. (5 x 1)( x 2)

2 x 1

2x

4 2x2

8. 3x( x 1)

2 x 2 3x 7

10. (2 x 3)(3x 1) 7 x 12.

4

13. (2 x 3)2 8 x

5

2

x 4

x 1

14. (3x 4)2

4

x

Solutions to the odd-numbered problems and answers to the even-numbered problems: 1. 3x2 5x 12 0 a 3, b 5, and c

Revised 04/10

2. x 12

5

3 or x 2

7 4

x x

x

b

b 2 4ac 2a

5

52 4(3)( 12) 2 3

5

169 6

x

5 13 6

x

5 13 or x 6

x

4 or x 3

5 13 6

3

3. x2 10 x 22

4. x

x 2 10 x 22 0 a 1, b

( 10)

x

x

10

22

b 2 4ac 2a

b

x

x

10, and c

( 10) 2 4(1)(22) 2(1)

12 2(1)

10 2 3 2(1)

x 5

3

Revised 04/10

6

1 i

5.

1 2 x 3

1 x 2

5 6

6 1 2 x 1 3

6. 5

6 1 x 1 2

2

3

6 1 2 x 1 3

6 1

6 5 1 6 1

1 x 2

6 1

1

1

5 6 1

2 x 2 3x 5 2 x 2 3x 5 0 a

x

x

x

x

2, b 3, and c

5

b

b 2 4ac 2a

3

( 3) 2 4(2)( 5) 2(2)

3

9 40 4

3

49 4

x

3 7 4

x

4 or x 4

x 1 or x

Revised 04/10

10 4 5 2 7

22

7. x2

4x2 2 x

x

8. x

3

89 10

3x 2 2 x 2 0 a

3, b

x

( 2) 2 4(3)( 2) 2(3)

( 2)

x

2

b 2 4ac 2a

b

x

x

2, and c

2

28 6

2 2 7 6 21

x

7 6

1

2 1 x

7 6 3

x

1

7 3

9. (5 x 1)( x 2)

2 x 2 3x 7

10. x

5 x 2 9 x 2 2 x 2 3x 7 3x 2 6 x 5 0 a 3, b 6, and c 5

x

x

b

b 2 4ac 2a

6

62 4(3)(5) 2(3)

Revised 04/10

8

2 2

6

x

24 6

6 2 6i 6

x

2

3

x

6i 6

1

2

3

x

6i 6 3

3

x

11.

6i 3

3x 2

2

12. x

4 , LCD is 2( x 1)

x 1

2( x 1) 3 x 1 2

2( x 1) 2 1 x 1

2( x 1)(4)

1

1

2 ( x 1) 3 x 1 2

2 ( x 1)

2

1

x 1

2( x 1)(4)

1

1

( x 1)(3x) 2(2)

2(4)( x 1)

3x 2 3x 4 8 x 8 3x 2 5 x 4 0 a 3, b

x

b

Revised 04/10

5, c

4

b 2 4ac 2a

9

17

15 i 8

x

x

( 5) 2 4(3)( 4) 2(3)

( 5)

x

5

25 48 6

5

73 6

13. (2 x 3)2 8 x

14. x

4 x 2 12 x 9 8 x 4 x 2 20 x 9 0 a

x

x

x

4, b

20, and c 9 b 2 4ac 2a

b

( 20) 2 4(4)(9) 2(4)

( 20)

20

256 8

x

20 16 8

x

20 16 or x 8

x

9 or x 2

Revised 04/10

20 16 8

1 2

10

16 or x 1 9

NOTE: You can get additional instruction and practice by going to the following web sites: http://www.purplemath.com/modules/solvquad4.htm This website has several worked out examples. It also shows the connection between the x-intercepts of a quadratic function and the solution to the related quadratic equation. http://www.purplemath.com/modules/quadform.htm This website has several examples with explanation. http://webmath.com/quadform.html This website has several interactive examples. http://www.youtube.com/watch?v=EeVqtpuMFOU This website has several video examples. http://www.sosmath.com/algebra/factor/fac08/fac08.html This website has the derivation of the quadratic formula, worked out examples, and practice problems. http://hotmath.com/help/gt/genericalg1/section_10_3.html?problem=0#anchor _0 This website has many interactive examples.

Revised 04/10

11