Completing the Square
Solving Quadratic Equations
Overview
Before – you solved equations by finding square roots
Now – you will solve quadratic equations by completing the square
Why? – so you can calculate a baseball’s maximum height
Solve a quadratic equation by finding square roots
Solve x2 + 20x + 100 = 81
x = -19 or -1
x2 + 20x + 100 = 81 | Write original equation. |
(x + 10)2 = 81 | Write the left side as a binomial squared. |
x + 10 = ± 9 | Take square root of each side |
x = -10 ± 9 | Solve for x. |
Make a perfect square trinomial
Find the value of c that makes x2 – 26x + c a perfect square trinomial. Then write the expression as the square of a binomial.
Step 1
Find half the coefficient of x.
-26 ÷ 2 = -13
Step 2
Square the result of Step 1.
-132 = 169
Step 3
Replace c with the result of Step 2.
X2 – 26x + 169
The trinomial x2 – 26x + c is a perfect square trinomial when c = 169.
Then x2 – 26x + 169 = (x – 13)(x – 13) = (x – 13)2
Solve ax2 + bx + c = 0 when a = 1
Solve x2 – 10x + 1 = 0
The solutions are 5 + 2√6 and 5 - 2√6.
x2 – 10x + 1 = 0
Write original equation.
x2 – 10x = -1
Write left side in the form of x2 + bx.
x2 – 10x + 25 = -1 + 25
Add (-10/2)2 = (-5)2 = 25 to both sides.
(x – 5)2 = 24
Write left side as a binomial squared.
x – 5 = ± √24
Take square roots of each side.
x = 5 ± √24
Solve for x.
x = 5 ± 2√6
Simplify √24 = √4 * √6 = 2√6
Solve ax2 + bx + c = 0 when a ≠ 1
Solve 2x2 + 8x + 14 = 0
The solutions are -2 + i√3 and -2 - i√3.
2x2 + 8x + 14 = 0
Write original equation.
x2 + 4x + 7 = 0
Divide each side by the coefficient of x2.
x2 +4x = -7
Write left side in the form of x2 + bx.
x2 +4x + 4 = -7 + 4
Add (4/2)2 = (2)2 = 4 to both sides.
(x + 2)2 = -3
Write left side as a binomial squared.
x + 2 = ± √-3
Take square roots of each side.
x = -2 ± √-3
Solve for x.
x = -2 ± i√3
Write in terms of the imaginary unit i.
Write a quadratic function in vertex form
Write y = x2 – 10x + 22 in vertex form. Then identify the vertex.
The vertex form of the function is y = (x - 5)2 – 3. The vertex is (5, -3).
y = x2 – 10x + 22
Write original equation.
y + ? = (x2 -10x + ? ) + 22
Prepare to complete the square.
y + 25 = (x2 -10x + 25) + 22
Add (10/2)2 = (-5)2 = 25 to both sides.
y + 25 = (x - 5)2 + 22
Write x2 – 10x + 25 as a binomial squared.
y = (x - 5)2 – 3
Solve for y.
Find the maximum value of a quadratic function
The height y (in feet) of a baseball t seconds after it is hit is given by this function:
y = -16t2 + 96t + 3
Find the maximum height of the baseball.
Solution: The maximum height of the baseball is the y-coordinate of the vertex of the parabola with the given equation.
Find the maximum value of a quadratic function (cont.)
How do we solve this function?
The vertex is (3, 147). So, the maximum height of the baseball is 147 feet.
y = -16t2 – 96t + 3
Write original equation.
y + ? = -16(t2 – 6t) + 3
Factor -16 from the first two terms.
y + -16( ? ) = -16(t2 - 6t+ ? ) + 3
Prepare to complete the square.
y + -16(9) = (t2 - 6x + 9) + 3
Add-16(9) to both sides.
y - 144 = (x - 3)2 + 3
Write t2 – 6t + 9 as a binomial squared.
y = (x - 3)2 + 147
Solve for y.
References
Larson, R, et al. (2011). Algebra 2. Orlando, Florida: Houghton Mifflin Harcourt Publishing Company.