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Mastering Quadratics Through Completing the Square

Learn how to find the vertex and 'A' of quadratic equations through completing the square method. Practice solving quadratics and transforming equations from standard to vertex form.

jaquelyn-albert
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Mastering Quadratics Through Completing the Square

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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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