advertisement

Completing the Square Factoring review The factoring that you have learnt thus far allows you to ﬁnd the zeros of a quadra9c func9on (AKA second degree polynomial) Completing the square is another method used to solve quadratic equations. It allows you to identify the vertex of a parabola when graphing. The general form of a quadratic equation is: 2 ax + bx + c = 0, a≠0 The factored form of a quadratic equation is: a(x + x1)(x + x2)= 0, a≠0 Where x1 and x2 are the zeros of the function The standard form of a quadratic equation is: a(x – 2 h) + k = 0, a≠0 Where (h,k) is the vertex of the function Step 1 Write the equation in the form 2 ax + bx ___ +c = 0 Step 2 If a ≠ 1, divide the first two term of the equation by a. Step 3 Complete the Square – Take ½ of b, and square the result. – Add and subtract this result to the equation Step 4 2 (b/a) Remove from the brackets (remember to multiply it by a) Step 5 Factor the trinomial left in the brackets as a perfect square Step 6 Simplify the term(s) left outside of the brackets Comple9ng the square f(x) = ax2 + bx + c f(x) = x2 -6x + 8 1. factor out a from the first two terms ONLY 2. complete the square of the first two terms 3. write your answer in standard form Comple9ng the square f(x) = ax2 + bx + c f(x) = x2 -6x + 8 f(x) = x2 -6x + 8 f(x) = (x2 -6x )+8 1. factor out a 2. complete the square 3. re-write in standard form Comple9ng the square f(x) = ax2 + bx + c f(x) = x2 -6x + 8 f(x) = x2 -6x + 8 f(x) = (x2 -6x )+8 f(x) = (x2 -6x +9 - 9) + 8 f(x) = (x2 -6x +9) -9 + 8 f(x) = (x2 -6x +9) - 1 1. factor out a 2. complete the square 3. re-write in standard form Comple9ng the square f(x) = ax2 + bx + c f(x) = x2 -6x + 8 f(x) = x2 -6x + 8 f(x) = (x2 -6x )+8 f(x) = (x2 -6x +9 - 9) + 8 f(x) = (x2 -6x +9) -9 + 8 f(x) = (x2 -6x +9) - 1 f(x) = (x -3)2 - 1 1. factor out a 2. complete the square 3. re-write in standard form Complete the Square Examples 1. x2 + 5x + 6 8. x2 + 13x + 12 2. x2 − 6x + 4 9. x2 + 8x + 12 3. x2 + 10x – 24 10. x2 − 8x – 9 4. x2 + 3x – 1 11. x2 + 13x – 48 5. x2 + 6x + 8 12. x2 − 20x + 64 6. x2 + 4x – 5 13. x2 − 7x – 4 7. x2 − 7x + 10 14. x2 − 2x − 3 Answers Comple9ng the square – Diﬃcult f(x) = ax2 + bx + c f(x) = 4x2 + 8x -60 1. factor out a from the first two terms ONLY 2. complete the square of the first two terms 3. write your answer in standard form Comple9ng the square f(x) = ax2 + bx + c f(x) = 4x2 + 8x -60 f(x) = 4x2 + 8x -60 f(x) = 4(x2 + 2x) - 60 1. factor out a 2. complete the square 3. re-write in standard form Comple9ng the square f(x) = ax2 + bx + c f(x) = 4x2 + 8x -60 2 f(x) = 4x + 8x -60 2 f(x) = 4(x + 2x) -60 2 f(x) = 4(x2 + 2x +2 -2) -60 f(x) = 4(x + 2x +2) -8 -60 1. factor out a 2. complete the square 3. re-write in standard form Comple9ng the square f(x) = ax2 + bx + c f(x) = 4x2 + 8x -60 2 f(x) = 4x + 8x -60 2 f(x) = 4(x + 2x) -60 2 f(x) = 4(x 2 + 2x +2 -2) -60 f(x) = 4(x + 2x +2) -8 -60 f(x) = 4(x + 1) 2 -68 1. factor out a 2. complete the square 3. re-write in standard form