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3. (x + 4) 2 x+ 4 x +4+4 x2x2 4x 16 Completing The Square Some quadratic functions can written as a perfect squares. x 2 + 8x + 16x 2 + 10x + 25 (x + 5) 2 x+ 5 x 5x 25 +5+5 5x x2x2 We can show this geometrically when the coefficient of x is positive. When we write expressions in this form it is known as completing the square.

4. Completing The Square Some quadratic functions can written as a perfect square. x 2 + 8x + 16x 2 + 10x + 25 (x + 5) 2 (x + 4) 2 (x - 2) 2 (x - 6) 2 x 2 - 4x + 4x 2 - 12x + 36 Similarly when the coefficient of x is negative: What is the relationship between the constant term and the coefficient of x? The constant term is always (half the coefficient of x) 2.

5. Completing The Square x 2 + 3x + 2.25x 2 + 5x + 6.25 (x + 2.5) 2 (x + 1.5) 2 (x - 3.5) 2 (x - 4.5) 2 x 2 - 7x + 12.25x 2 - 9x + 20.25 When the coefficient of x is odd we can still write a quadratic expression as a non-perfect square, provided that the constant term is (half the coefficient of x) 2

6. = (x + 5) 2 = (x + 2) 2 = (x - 3) 2 = (x - 6) 2 x 2 + 4x x 2 + 10x x 2 - 6x x 2 - 12x Completing The Square This method enables us to write equivalent expressions for quadratics of the form ax 2 + bx. We simply half the coefficient of x to complete the square then remember to correct for the constant term. - 4 - 25 - 9 - 36

7. = (x + 2.5) 2 = (x + 1.5) 2 = (x - 3.5) 2 = (x - 4.5) 2 x 2 + 3x x 2 + 5x x 2 - 7x x 2 - 9x Completing The Square This method enables us to write equivalent expressions for quadratics of the form ax 2 + bx. We simply half the coefficient of x to complete the square then remember to correct for the constant term. - 2.25 - 6.25 - 12.25 - 20.25

8. = (x + 5) 2 = (x + 2) 2 = (x - 1) 2 = (x - 6) 2 x 2 + 4x + 3 x 2 + 10x + 15 x 2 - 2x + 10 x 2 - 12x - 1 Completing The Square We can also write equivalent expressions for quadratics of the form ax 2 + bx + c. Again, we simply half the coefficient of x to complete the square and remember to take extra care in correcting for the constant term. - 1 - 10 + 9 - 37

9. = (x + 1) 2 = (x + 3) 2 = (x - 1.5) 2 = (x - 2.5) 2 x 2 + 6x - 8x 2 + 2x + 9 x 2 - 3x + 2 x 2 - 5x - 3 Completing The Square We can also write equivalent expressions for quadratics of the form ax 2 + bx + c. Again, we simply half the coefficient of x to complete the square and remember to take extra care in correcting for the constant term. - 17 + 8 - 0.25 - 9.25 Now try these

10. Questions 1 Completing The Square We can also write equivalent expressions for quadratics of the form ax 2 + bx + c. Again, we simply half the coefficient of x to complete the square and remember to take extra care in correcting for the constant term. Questions: Write the following in completed square form: 1. x 2 + 8x + 10 2. x 2 - 6x + 1 3. x 2 - 2x + 2 4. x 2 + 10x + 30 5. x 2 + 6x - 5 6. x 2 - 12x - 3 7. x 2 - x + 4 8. x 2 - 3x + 7.25 = (x + 4) 2 - 6 = (x - 3) 2 - 8 = (x - 1) 2 + 1 = (x + 5) 2 + 5 = (x + 3) 2 - 14 = (x - 6) 2 - 39 = (x - ½) 2 + 3.75 = (x - 1.5) 2 + 5

11. Solving Quadratic Equations by Completing the Square Example Question 1: Solve x 2 + 4x - 6 = 0 (to 2 dp) x 2 + 4x - 6 = 0 x 2 + 4x = 6 (x + 2) 2 - 4 = 6 (x + 2) 2 = 10 x + 2 = +/-  10 x = - 2 +/-  10 Re-arrange Complete the square  Both sides Re-arrange x = 1.16 and - 5.16 Re-arrange and solve It is often more efficient to solve quadratic equations by completing the square rather than using the common formula. This is particularly true when the coefficient of x 2 is 1. The process of completing the square gives more insight into the mathematics behind the solution than does the formula.

12. Example Question 2: Solve x 2 + 6x + 3 = 0 (to 2 dp) x 2 + 6x + 3 = 0 x 2 + 6x = - 3 (x + 3) 2 - 9 = - 3 (x + 3) 2 = 6 x + 3 = +/-  6 x = - 3 +/-  6 Re-arrange Complete the square  Both sides Re-arrange x = - 0.55 and - 5.45 Re-arrange and solve Solving Quadratic Equations by Completing the Square It is often more efficient to solve quadratic equations by completing the square rather than using the common formula. This is particularly true when the coefficient of x 2 is 1. The process of completing the square gives more insight into the mathematics behind the solution than does the formula.

13. Example Question 3: Solve x 2 + 8x + 6 = - 7 (to 2 dp) x 2 + 8x + 6 = - 7 x 2 + 8x = - 13 (x + 4) 2 - 16 = - 13 (x + 4) 2 = 3 x + 4 = +/-  3 x = - 4 +/-  3 Re-arrange Complete the square  Both sides Re-arrange x = - 2.27 and - 5.73 Re-arrange and solve Solving Quadratic Equations by Completing the Square It is often more efficient to solve quadratic equations by completing the square rather than using the common formula. This is particularly true when the coefficient of x 2 is 1. The process of completing the square gives more insight into the mathematics behind the solution than does the formula.

14. Questions 2 1. x 2 + 6x + 1 = 0 2. x 2 - 8x + 3 = 0 3. x 2 - 4x - 7 = 3 4. x 2 + 3x + 1= 0 x = - 0.17 and - 5.83 Solving Quadratic Equations by Completing the Square It is often more efficient to solve quadratic equations by completing the square rather than using the common formula. This is particularly true when the coefficient of x 2 is 1. The process of completing the square gives more insight into the mathematics behind the solution than does the formula. Re-arrange Complete the square  Both sides Re-arrange Re-arrange and solve x = 7.61 and 0.39 x = 5.74 and - 1.74 x = - 0.38 and - 2.62 Questions: Solve the following by completing the square (2 dp):