Algebra 9.5 Solving Quadratic Equations Using the Quadratic Formula This is an important section as...

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.