4.6 Completing the Square

Objectives: To solve equations by completing the square. To rewrite

functions by completing the square.

I. Solutions of Quadratic Equation: As we saw in the Solving by factoring section the solutions

of a quadratic are the x- intercepts, called the zeros. They can also be referred to as solution or roots.

What do you think the graph of the quadratic look like if there was: A. One Real Solution B. Two Real Solution

C. No Real Solution

Solutions that are not real are referred to as imaginary or complex roots. They are found when taking an even root of a negative.

II. Simplify the following roots.A. B.

C.

4

7

12

i 1

III. Solve by Finding Square Roots.Steps:1. Isolate the term with the variables on one side

of the equation.

2. Divide both sides by the coefficient a.

3. Take the square root of both sides.

4. Remember when taking square roots, it is positive and negative value

Solve the equations. A. B.

C. D. x² + 10x +25 = 12

23 192x 2120 2

2x

235 32

4x

Think about x² + 10x +25, what is the relationship between the c and b terms.

The c term is . Thus, making it a perfect square trinomial.

What value completes the square for x² + 8x?

2

2

b

IV. Solving an equation by Completing the SquareSteps:1. Rewrite the equation in the form of x²+bx=c. To

do this, get all terms with the variable on one side of the equation and the constant on the other side. Divide all the terms of the equation by the coefficient of x² if it is not 1.

2. Complete the square by adding to each side of the equation.

3. Factor the trinomial.4. Find the square roots.5. Solve for x.

2

2

b

A. B. 2 6 7 0x x 2 2 2 0x x

C. D. 2x² - x + 3 = x+9 24 6 1 0x x

V. You can complete the square to change a quadratic function to vertex form?

y = x² +4x -6y = x² +4x +____ -6 - _____ (Add to complete

the square. Also, subtract to leave the

function unchanged.)y= (x+ ___) ² -6 - 2² Factor the perfect sq

trinomial.Y= (x + ___)² - _____ Simplify.The vertex is ( , ) The y intercept is ____.

2

2

b

Write the following in vertex form.A. y= x²+3x-6

B. y= 2x²-6x-1

• Homework• Pre-AP• p. 237 # 13-61 odd, 75