Algebra 1 Common Core Student Edition, Grade 8-9

Here's W hy It W orks If you complete the square for the general equation

ax2 + bx + c = 0, you can derive the quadratic formula.

Step 1 Write ax2 + bx + c = 0 so the coefficient of x2 is 1.

ax2 + bx + c = 0

x + a x + a = 02 i b , c „ Divide each side by a.

Step 2 Complete the square

x +ax =.-2

x2 + K +

This step uses th e

property

which you w ill study

in Lesson 10-2.

_£+m 2

a (2a J

— £ fa2

a 4 a2

4ac , b 2

4 a2 4 a2

b\2 _ b2 - 4 ac

4 a2

Step 3 Solve the equation for x.

V> + a)2=±V/ s^

Vfo2 - 4ac

2 a

[IT -

(*+ £)’ -

(*+i )

Subtract § from each side.

Add f^)2 to each side.

Write the left side as a square.

Multiply -§ by jz to get like denom inators.

Simplify the right side.

Take square roots o f each side.

Simplify the right side.

b_ _, Vfo2 — 4 a c

2a ~ 2 a

x = -b ± Vfo2 2a - 4qc

Subtract ^ from each side.

Simplify.

Thi nk
Why do you need to
write the equation in
standard form?
You can o n ly use th e
quadratic formula w ith
eq u a tion s in th e fo rm
ax 2 + bx + c = 0.

Be sure to write a quadratic equation in standard form before using the quadratic

formula.

Usi ng t he Qu ad r at i c For mul a

What are the solutions of x 2 - 8 = 2x? Use the quadratic formula.

Write the equation in standard form.

Use th e quadratic form ula.

xz — 2x - 8 = 0

x = '—b ± Vfo2 2 a — 4 a c

x =

-(-2 ) ± V (“ 2)2 -4 (l)(—8)

2 ( 1 )

X =

X =

x = 4

2 ± V36

2

2 + (^6) or
or
X _^2 - 26
x = —2
Substitute 1 for a, - 2 for b, and -8 for c.
Simplify.
Write as two equations.
Simplify.
Go t It? 1. What are the solutions of x2 - 4x = 21? Use the quadratic formula.
SHI | Lesso n 9- 6 Th e Q u ad r at ic Fo rm u la an d t h e Discrim in an t^583