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3.7 Warm Up

3.7 Warm Up. Find the vertex and A of S. 1. y = (x – 2) ² - 6 2. y = (x + 5) ² + 6 3. y = (x – 8) ² - 2 4. y = 2(x – 4)(x – 6) 5. y = -(x + 3)(x – 5). 3.7 Complete the Square. 2 Reasons to Complete the Square. To solve quadratics To write the function from standard to vertex form.

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3.7 Warm Up

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  1. 3.7 Warm Up Find the vertex and A of S. 1. y = (x – 2)² - 6 2. y = (x + 5)² + 6 3. y = (x – 8)² - 2 4. y = 2(x – 4)(x – 6) 5. y = -(x + 3)(x – 5)

  2. 3.7 Complete the Square

  3. 2 Reasons to Complete the Square • To solve quadratics • To write the function from standard to vertex form.

  4. ANSWER The solutions are 4 + 5 = 9 and 4 –5 = – 1. EXAMPLE 1 Solve a quadratic equation by finding square roots Solve x2 – 8x + 16 = 25. Write original equation. x2 – 8x + 16 = 25 (x – 4)2 = 25 Write left side as a binomial squared. x – 4 = +5 Take square roots of each side. x = 4 + 5 Solve for x.

  5. GUIDED PRACTICE Solve the equation by finding square roots. 1. x2 + 6x + 9 = 36. ANSWER 3 and –9. 2. x2 – 10x + 25 = 1. ANSWER 4 and 6. 3. x2 – 24x + 144 = 100. ANSWER 2 and 22.

  6. 81 9 4 2 for Examples 1 and 2 GUIDED PRACTICE Find the value of c that makes the expression a perfect square trinomial.Then write the expression as the square of a binomial. x2 + 14x + c 4. ANSWER 49 ; (x + 7)2 x2 + 22x + c 5. ANSWER 121 ; (x + 11)2 x2 – 9x + c 6. ; (x – )2. ANSWER

  7. To solve quadratics by completing the square. . . • Write one side of the equation in the form x2 + bx (move the c over) • Find the term to complete the square and add to both sides • When you add (b/2)2, you now can factor it into • Then, take the square root to solve.

  8. ) ( –12 Add to each side. 2 36 (–6) 2 = = 2 x – 6 = + 32 x = 6 + 32 x = 6 + 4 2 2 Simplify: 32 16 2 4 = = ANSWER 2 2 The solutions are 6 + 4 and6 – 4 EXAMPLE 3 Solve ax2 + bx + c = 0 when a = 1 Solve x2 – 12x + 4 = 0 by completing the square. x2 – 12x + 4 = 0 Write original equation. x2 – 12x = –4 Write left side in the form x2 + bx. x2– 12x + 36 = –4 + 36 (x – 6)2 = 32 Write left side as a binomial squared. Take square roots of each side. Solve for x.

  9. Solve ax2 + bx + c = 0 when a = 1 The solutions are –2 + i and–2 – i 3 3 . ) ( 4 Add to each side. 2 4 2 2 = = 2 x + 2 = + –3 x = –2 + –3 x = –2 +i 3 EXAMPLE 4 Solve 2x2 + 8x + 14 = 0 by completing the square. 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 x2 + bx. x2+ 4x + 4 = –7 + 4 (x + 2)2 = –3 Write left side as a binomial squared. Take square roots of each side. Solve for x. Write in terms of the imaginary unit i.

  10. ANSWER –3+ 5 ANSWER –2 + 10 ANSWER 5 + 17 ANSWER –4 +3 2 ANSWER ANSWER 1 + 2 2 1 + 26 GUIDED PRACTICE Solve the equation by completing the square. 7. 10. 3x2 + 12x – 18 = 0 x2 + 6x + 4 = 0 8. x2 – 10x + 8 = 0 11. 6x(x + 8) = 12 9. 12. 4p(p – 2) = 100 2n2 – 4n – 14 = 0

  11. ) ( –10 Add to each side. 2 25 (–5) 2 = = 2 ANSWER The vertex form of the function is y = (x – 5)2– 3. The vertex is (5, –3). EXAMPLE 6 Write a quadratic function in vertex form Write y = x2 – 10x + 22 in vertex form. Then identify the vertex. y = x2 – 10x + 22 Write original function. y + ?= (x2–10x + ?) + 22 Prepare to complete the square. y + 25= (x2– 10x + 25) + 22 y + 25 = (x – 5)2 + 22 Write x2 – 10x + 25 as a binomial squared. y = (x – 5)2– 3 Solve for y.

  12. GUIDED PRACTICE Write the quadratic function in vertex form. Then identify the vertex. 13. y = x2 – 8x + 17 y = (x – 4)2+ 1 ; (4, 1). ANSWER 14. y = x2 + 6x + 3 y = (x + 3)2– 6 ; (–3, –6) ANSWER 15. f(x) = x2 – 4x – 4 ANSWER y = (x – 2)2– 8 ; (2 , –8)

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