Algebra

9.5 Solving Quadratic Equations Using the Quadratic Formula

This is an important section as there are many questions on the STAR test about the quadratic formula.

Solve

2x2 + 10 = 28

- 10 -10

2x2 = 18

x2 = 9 x = + 3 These are the solutions/roots of the equation.

We did not need the quadratic formula to solve this quadratic equation because it was in the form…

Ax2 + C = # where b = 0.

What is the quadratic formula?

It is a formula used to solve any quadratic equation in the form…

ax2 + bx + c = 0 when a ≠ 0 and

b2 – 4ac ≥ 0.

Using the formula will produce the solutions(roots) of the equation.

Here it is…try to memorize it…

x =

This formula can be used to find the roots of any quadratic equation in the form ax2 + bx + c = 0.

b2 – 4ac

2a

-b +

http://www.mathmadness.org/resources/Quadratic+Formula.mp3

Quadratic+Formula.mp3

Find the roots of 2x2 + 10 = 28 using the quadratic formula…

The equation must be in the form ax2 + bx + c = 0 before using the quadratic formula.

2x2 + 10 = 28 -28 -28 2x2 - 18 = 0

Remember the roots were x = + 3

Must be 0 in order to use the quadratic formula.

-b + b2 – 4ac

2aX =

-0 + (0)2 – 4(2)(-18)

2(2)X =

+ 144

4X =

+ 12

4X =

3 and - 3X =

a = 2, b = 0, c = -18

These are the roots of the equation.

Find the roots of -3x2 + 4x = -5 using the quadratic formula…

-3x2 + 4x = -5 +5 +5 -3x2 + 4x + 5 = 0

-b + b2 – 4ac

2aX =

-4 + (4)2 – 4(-3)(5)

2(-3)X =

a = -3, b = 4, c = 5

-4 + 16 + 60

2(-3)X =

-4 + 76

-6X =

76

4 19

-4 + 2 19

-6X =

Can you reduce?

Both numbers in the numerator must have common factorsof the denominator.

2 + 19

3X =These are

the two roots.

and 2 - 19

3

Yes, by -2.

You try! Find the roots of 4x2 - x = 7 using the quadratic formula… who can do it on the board?

4x2 - x = 7 -7 -7 4x2 - x - 7 = 0

-b + b2 – 4ac

2aX =

1 + (-1)2 – 4(4)(-7)

2(4)X =

a = 4, b = -1, c = -7

1 + 1 + 112

8X =

1 + 113

8X =

These arethe two roots of the equation.

1 - 113

8

and

Can you reduce?

No.

Note: This is on the STAR test.

The roots of a quadratic equation are the x-intercepts of the graph of the quadratic (parabola).

The roots = x-intercepts

You try! Find the x-intercepts of the graph of y = x2 + 5x – 6.

using the quadratic formula… who can do it on the board?

y = x2 + 5x – 6 0 = x2 + 5x – 6 (substitute 0 for y)

-b + b2 – 4ac

2aX =

-5 + (5)2 – 4(1)(-6)

2(1)X =

a = 1, b = 5, c = -6

-5 + 49

2X =

-5 + 7

2X =

These arethe two roots and x-intercepts.

-5 - 7

2

and

2

2X =

-12

2

and

1X = -6 and

One from the HW

P. 536 #46

HW

P. 536-537 # 33-36, 42-46, 53-55

Leave answers in simplified radical form.