Why we complete the square  We have learned how to factor quadratic expressions to solve.  Many quadratic equations contain expressions that cannot be.

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Presentation transcript:

Why we complete the square  We have learned how to factor quadratic expressions to solve.  Many quadratic equations contain expressions that cannot be easily factored.  For equations containing these types of expressions, you can use square roots to find roots.

If a quadratic expression of the form x 2 + bx + c is not a perfect square trinomial, you can add a term to form a perfect square trinomial. This is called completing the square.

Assignment #41  Pg. 237, #13-17odd, 18, 28-33

DO NOW Get out your laptop and a pencil.

You can complete the square to solve quadratic equations. Completing the Square

Complete the square for the expression. Write the resulting expression as a binomial squared. x 2 + 4x + Add. Factor. Find. x 2 + 4x + 4 (x + 2) 2 Check Find the square of the binomial. = x 2 + 4x + 4 (x + 2) 2 = (x + 2)(x + 2) Completing The Square: 1

Complete the square for the expression. Write the resulting expression as a binomial squared. x 2 + 3x + Add. Factor. Find. Check Find the square of the binomial. Completing The Square: 2

Solve the equation by completing the square. x 2 = 12x – 20 x 2 – 12x = –20 Collect variable terms on one side. Simplify. Set up to complete the square. x 2 – 12x + = –20 + Add to both sides. x 2 – 12x + 36 = – Completing The Square To Find Roots: 1

(x – 6) 2 = 16 Factor. Take the square root of both sides. x – 6 = ±4 Solve for x. x – 6 = 4 or x – 6 = –4 Simplify. x = 10 or x = 2 Completing The Square To Find Roots: 1

Vertex Form The result of completing the square and factoring was: (x – 6) 2 = 16 If we subtract 16 from both sides, we end up with our function in vertex form. f(x) = (x – 6)

Solve the equation by completing the square. 18x + 3x 2 = 45 x 2 + 6x = 15 Divide both sides by 3. Simplify. x 2 + 6x + = 15 + Add to both sides. x 2 + 6x + 9 = Set up to complete the square. Completing The Square To Find Roots: 2

Take the square root of both sides. Factor. Simplify. (x + 3) 2 = 24 Completing The Square To Find Roots: 2

3x 2 – 24x = 27 Solve the equation by completing the square. Divide both sides by 3. Simplify. Add to both sides. Set up to complete the square. x 2 – 8x = 9 x 2 –8x + = 9 + Completing The Square To Find Roots: 3

Solve the equation by completing the square. Factor. Solve for x. Simplify. Take the square root of both sides. x =–1 or x = 9 x – 4 =–5 or x – 4 = 5 Completing The Square To Find Roots: 3

Assignment #42 Pg. 237, #19-27odd, 40-44even, 52