10.5 Solve Quadratic Equations by Completing the Square
Goal • Solve a quadratic equation by completing the square.
Warm-up:
Foil:
1) Foil: (x + 4) 2
2) The answer to #1 is in the form x 2 +bx+c:
a) What is the value of “b”?
b) If you take half of “b” and square it, what do you get?
c) What part of the trinomial does that match?
3) Foil: (x - 3) 2
4) The answer to #3 is in the form x 2 +bx+c:
a) What is the value of “b”?
b) If you take half of “b” and square it, what do you get?
c) What part of the trinomial does that match?
5) Factor: x 2 + 10x + 25
a) What is the value of “b” in the original trinomial?
b) How does the value of “b” relate to the number portion of the factors?
c) If you take half of “b” and square it, what do you get?
d) What part of the trinomial does that match?
5) Factor: x 2 - 18x + 81
a) What is the value of “b” in the original trinomial?
d) What part of the trinomial does that match?
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c) If you take half of “b” and square it, what do you get?
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b) How does the value of “b” relate to the number portion of the factors?
Completing the Square
Words To complete the square for the expression x2 + bx, add the square of half the coefficient of
the term bx.
⎛ b ⎞2 ⎛x + b ⎞ 2
2
Algebra x + bx +⎜ 2 ⎜ = ⎜ 2 ⎜
⎝
⎠
⎝ ⎠
Example 1 – Completing the square
Find the value of c that makes the expression x 2 + 8x + c a perfect square trinomial. Then write the
expression as the square of the binomial.
Step 1 Find the value of c. For the expression to be a perfect square trinomial, c needs to be the
square of half the coefficient of the term bx.
Since b = _____, c =
Step 2 Write the expression as a perfect square trinomial. Then write the expression as the square
of a binomial.
x 2 + 8x + c = x 2 + 8x +
=
Square of a binomial
Example 2 – Completing the square
Write the expression as a perfect square trinomial. Then write the expression as the square of a
binomial.
x 2 - 6x + c = x 2 - 6x +
=
Square of a binomial
Now you try it! Find the value of c that makes the expression a perfect square trinomial. Then
write the expression as the square of a binomial.
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2. x2 – 4x + c
2
1. x 2 + 10x + c
Example 3 Solve a quadratic equation by completing the square
Solve x2 – 16x = -15 by completing the square.
x2 – 16x = –15
x2 – 16x + _____ = –15 + _____
Write original equation.
Add (_____)2, or _____ , to each side.
________ = –15 + _____
Write left side as the square of a
binomial.
________ = _____
Simplify the right side
________ = _______
Take square roots of each side
x = ________
Add_____ to each side
The solutions of the equation are _____________ and ____________.
Example 4 Solve a quadratic equation in standard form by completing the square
Solve 2x2 + 20x - 8 = 0 by completing the square.
2x2 + 20x - 8 = 0
2x2 + 20x = _____
x2 + 10x = _____
x2 + 10x + _____ = 4 + _____
Write original equation.
Add______ to each side
Divide each side by _____
Add (_____)2 or _____ , to each side.
________ = _____
Write left side as the square of a binomial
and simplify the right side
________ = _____
Take square roots of each side.
x = _____________
Subtract _____ from each side
Put in calculator LAST!
Now you try it! Solve the equation by completing the square. Round your solutions to the
nearest hundredth, if necessary.
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4. 5s2 + 60s + 125 = 0
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3. r2 - 8r = 9