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Completing the Square Topic 7.2.1

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**Completing the Square 7.2.1 Topic California Standard:**

14.0 Students solve a quadratic equation by factoring or completing the square. What it means for you: You’ll form perfect square trinomials by adding numbers to binomial expressions. Key words: completing the square perfect square trinomial binomial

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**Completing the Square 7.2.1 Topic**

“Completing the square” is another method for solving quadratic equations — but before you solve any equations, you need to know how completing the square actually works.

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**Completing the Square 7.2.1 Topic**

Writing Perfect Square Trinomials as Perfect Squares An expression such as (x + 1)2 is called a perfect square — because it’s (something)2. In a similar way, an expression such as x2 + 2x + 1 is called a perfect square trinomial (“trinomial” because it has 3 terms). This is because it can be written as a perfect square: x2 + 2x + 1 = (x + 1)2 Any trinomial of the form x2 + 2dx + d2 is a perfect square trinomial, since it can be written as the square of a binomial: x2 + 2dx + d2 = (x + d)2

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**Completing the Square 7.2.1 Topic**

Converting Binomials to Perfect Squares The binomial expression x2 + 4x is not a perfect square — it can’t be written as the square of a binomial. However, it can be turned into a perfect square trinomial if you add a constant (a number) to the expression.

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**Completing the Square 7.2.1 Topic x2 + 2dx + d2 = (x + d)2**

Example 1 Convert x2 + 4x to a perfect square trinomial. Solution To do this you have to add a number to the original expression. First look at the form of perfect square trinomials, and compare the coefficient of x with the constant term (the number not followed by x or x2): x2 + 2dx + d2 = (x + d)2 The coefficient of x is 2d, while the constant term is d2. The constant term is the square of half of the coefficient of x. To convert x2 + 4x to a perfect square trinomial, add the square of half of 4 — that is, add 22 = 4, to give x2 + 4x + 4 = (x + 2)2 Solution follows…

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**Completing the Square 7.2.1 Topic Guided Practice**

By adding a constant, convert each of these binomials into a perfect square trinomial. 1. x2 + 14x 2. x2 – 12x 3. x2 + 2x 4. x2 – 8x 5. y2 + 20y 6. p2 – 16p x2 + 14x + 49 x2 – 12x + 36 x2 + 2x + 1 x2 – 8x + 16 y2 + 20y + 100 p2 – 16p + 64 Solution follows…

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**Completing the Square 7.2.1 Topic Completing the Square for x2 + bx**

To convert x2 + bx into a perfect square trinomial, add b 2 The resulting trinomial is x b 2

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**Completing the Square 7.2.1 Topic**

Example 2 Form a perfect square trinomial from x2 + 8x. Solution Here, b = 8, so to complete the square you add 2 8 This gives you x2 + 8x + 16. Solution follows…

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**Completing the Square 7.2.1 Topic**

Example 3 What must be added to y2 – 12y to make it a perfect square trinomial? Solution This time, b = –12 To complete the square you add = (–6)2 = 36. 2 –12 So 36 must be added (giving y2 – 12y + 36). Solution follows…

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**Completing the Square 7.2.1 Topic**

Example 4 Suppose x2 – 10x + c is a perfect square trinomial, and is equal to (x + k)2. What are the values of c and k? Solution Here the coefficient of x is –10. So to form a perfect square trinomial, the constant term has to be the square of half of –10, so c = (–5)2 = 25. Therefore x2 – 10x + c = x2 – 10x + 25 = (x + k)2. Now multiply out the parentheses of (x + k)2. (x + k)2 = x2 + 2kx + k2 Solution continues… Solution follows…

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**Completing the Square 7.2.1 Topic**

Example 4 Suppose x2 – 10x + c is a perfect square trinomial, and is equal to (x + k)2. What are the values of c and k? Solution (continued) (x + k)2 has to equal x2 – 10x + 25, which gives x2 – 10x + 25 = x2 + 2kx + k2. Equate the coefficients of x: the coefficient of x on the left-hand side is –10, while on the right-hand side it is 2k. So –10 = 2k, or k = –5. Comparing the constant terms in a similar way, you find that 25 = k2 , which is also satisfied by k = –5.

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**Completing the Square 7.2.1 Topic Guided Practice**

Find the value of k that will make each expression below a perfect square trinomial. 7. x2 – 7x + k 8. q2 + 5q + k 9. x2 + 6mx + k 10. d2 – 2md + k k = = 49 4 2 –7 k = = 25 4 2 5 k = = 9m2 2 6m k = = m2 2 2m Solution follows…

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**Completing the Square 7.2.1 Topic Guided Practice**

Form a perfect square trinomial from the following expressions by adding a suitable term. 11. x2 + 10x 12. x2 – 16x 13. y2 + 2y 14. x2 + bx 15. y2 – 18y 16. a2 – 2a 17. y2 + 12y 18. y2 + 36y Add a constant equal to (10 ÷ 2)2 Þ x2 + 10x + 25 Add a constant equal to (–16 ÷ 2)2 Þ x2 – 16x + 64 Add a constant equal to (2 ÷ 2)2 Þ y2 + 2y + 1 Add a constant equal to (b ÷ 2)2 Þ x2 + bx + b2 4 Add a constant equal to (–18 ÷ 2)2 Þ y2 – 18y + 81 Add a constant equal to (–2 ÷ 2)2 Þ a2 – 2a + 1 Add a constant equal to (12 ÷ 2)2 Þ y2 + 12y + 36 Add a constant equal to (36 ÷ 2)2 Þ y2 + 36y + 324 Solution follows…

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**Completing the Square 7.2.1 Topic Guided Practice**

Form a perfect square trinomial from the following expressions by adding a suitable term. 19. 6x + 9 20. 1 – 8x – 20y Add a squared term, so that ax2 + 6x + 9 = (bx + 3)2 Find a and b, ax2 + 6x + 9 = b2x2 + 6bx + 9 Comparing x terms shows b = 1 Þ a = b2 = 1 So the trinomial is x2 + 6x + 9 Add a squared term, so that ax2 – 8x + 1 = (bx – 1)2 Find a and b, ax2 – 8x + 1 = b2x2 – 2bx + 1 Comparing x terms shows b = 4 Þ a = b2 = 16 So the trinomial is 16x2 – 8x + 1 Add a squared term, so that ay2 – 20y + 25 = (by – 5)2 Find a and b, ay2 – 20y + 25 = b2y2 – 10by + 25 Comparing y terms shows b = 2 Þ a = b2 = 4 So the trinomial is 4y2 – 20y + 25 Solution follows…

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**Completing the Square 7.2.1 Topic Guided Practice**

Form a perfect square trinomial from the following expressions by adding a suitable term. 22. 4y2 + 4yb 23. 4a2 + 12ab 24. 9a2 + 16b2 Add a b2 term, so that 4y2 + 4yb + a2b2 = (2y + ab)2 Find a, 4y2 + 4yb + a2b2 = 4y2 + 4aby + a2b2 Comparing y terms shows a = 1 Þ a2 = 1 So the trinomial is 4y2 + 4yb + b2 Add a b2 term, so that 4a2 + 12ab + c2b2 = (2a + cb)2 Find c, 4a2 + 12ab + c2b2 = 4a2 + 4cab + c2b2 Comparing a terms shows c = 3 Þ b2 = 9 So the trinomial is 4a2 + 12ab + 9b2 Add an ab term, so that 9a2 + cab + 16b2 = (3a + 4b)2 Find c, 9a2 + cab + 16b2 = 9a2 + 24ab + 16b2 Comparing ab terms shows c = 24 So the trinomial is 9a2 + 24ab + 16b2 Solution follows…

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**Completing the Square 7.2.1 Topic Guided Practice**

The quadratics below are perfect square trinomials. Find the value of c and k to make each statement true. 25. x2 – 6x + c = (x + k)2 26. x2 + 16x + c = (x + k)2 27. 4x2 + 12x + c = (2x + k)2 28. 9x2 + 30x + c = (3x + k)2 29. 4a2 – 4ab + cb2 = (2a + kb)2 30. 9a2 – 12ab + cb2 = (3a + kb)2 c = 9, k = –3 c = 64, k = 8 c = 9, k = 3 c = 25, k = 5 c = 1, k = –1 c = 4, k = –2 Solution follows…

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**Completing the Square 7.2.1 Topic**

If the Coefficient of x2 isn’t 1, Add a Number With an expression of the form ax2 + bx, you can add a number to make an expression of the form a(x + k)2.

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**Completing the Square 7.2.1 Topic**

Example 5 If 3x2 – 12x + m is equal to 3(x + d)2, what is m? Solution Multiply out the parentheses of 3(x + d)2 to get: 3(x + d)2 = 3x2 + 6dx + 3d2 So 3x2 – 12x + m = 3x2 + 6dx + 3d2 Equate the coefficients of x, and the constants, to get: –12 = 6d and m = 3d2 The first equation tells you that d = –2. And the second tells you that m = 3(–2)2, or m = 12. So 3x2 – 12x + 12 = 3(x – 2)2. Solution follows…

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**Completing the Square 7.2.1 Topic Completing the square for ax2 + bx**

The expression ax2 + bx can be changed to a trinomial of the form a(x + k)2. b 2 To do this, add = . 1 a b2 4a The resulting trinomial is a b 2a 2 x +

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**Completing the Square 7.2.1 Topic**

Example 6 Convert 2x2 + 10x to a perfect square trinomial. Solution 2 Here, a = 2 and b = 10, so you add = = 10 1 100 4 25 This gives you 2x2 + 10x 25 2 Solution follows…

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**Completing the Square 7.2.1 Topic Guided Practice**

The quadratics below are of the form a(x + d)2. Find the value of m and d in each equation. 31. 5x2 + 10x + m = 5(x + d)2 32. 4x2 – 24x + m = 4(x + d) x2 – 28x + m = 2(x + d)2 34. 3x2 – 30x + m = 3(x + d) x2 + 32x + m = 4(x + d)2 m = 5, d = 1 m = 36, d = –3 m = 98, d = –7 m = 75, d = –5 m = 64, d = 4 Solution follows…

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**Completing the Square 7.2.1 Topic Guided Practice**

The quadratics below are of the form a(x + d)2. Find the value of m and d in each equation. 36. 20x2 + 60x + m = 5(2x + d) x2 – 20x + m = 5(2x + d)2 38. 27x2 + 18x + m = 3(3x + d) x2 + 36x + m = 3(3x + d)2 40. 16x2 – 80x + m = 4(2x + d)2 m = 45, d = 3 m = 5, d = –1 m = 3, d = 1 m = 12, d = 2 m = 100, d = –5 Solution follows…

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**Completing the Square 7.2.1 Topic Guided Practice**

Add a term to convert each of the following into an expression of the form a(x + k)2. 41. 2x2 – 12x 42. 3a2 + 12a 43. 6y2 – 60y 44. 4x2 – 48x 45. 5x 46. 8x x + 12 x + 100 2x2 – 12x + 18 = 2(x – 3)2 3a2 + 12a + 12 = 3(a + 2)2 6y2 – 60y =6(y – 5)2 4x2 – 48x = 4(x – 6)2 5x2 + 70x = 5(x + 7)2 8x2 + 8x + 2 = 8(x + )2 1 2 27x2 + 63x + 12 = 27(x + )2 2 3 36x x = 36(x + )2 5 3 Solution follows…

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**Completing the Square 7.2.1 Topic Independent Practice**

Find the value of c that will make each expression below a perfect square trinomial. 1. x2 + 9x + c 2. x2 – 11x + c 3. x2 + 12xy + c 4. x2 – 10xy + c 81 4 121 4 36y2 25y2 Solution follows…

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**Completing the Square 7.2.1 Topic Independent Practice**

Complete the square for each quadratic expression below. 5. x2 – 6x 6. a2 – 14a 7. b2 – 10b 8. x2 + 8xy 9. c2 – 12bc 10. x2 + 4xy x2 – 6x + 9 a2 – 14a + 49 b2 – 10b + 25 x2 + 8xy + 16y2 c2 – 12bc + 36b2 x2 + 4xy + 4y2 Solution follows…

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**Completing the Square 7.2.1 Topic Independent Practice**

Find the value of m and d in each of the following. 11. 5x2 – 40x + m = 5(x + d)2 12. 2x2 + 20x + m = 2(x + d) x2 – 6x + m = 3(x + d)2 14. 3x2 – 30x + m = 3(x + d) x2 + 24x + m = 4(x + d)2 16. 7x2 – 28x + m = 7(x + d)2 m = 80, d = –4 m = 50, d = 5 m = 3, d = –1 m = 75, d = –5 m = 36, d = 3 m = 28, d = –2 Solution follows…

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**Completing the Square 7.2.1 Topic Independent Practice**

Add a term to convert each of the following into an expression of the form a(x + k)2. 17. 3x2 – 30x 18. 2x2 + 8x x2 – 48x 20. 5x x2 + 12 22. 36x 3(x – 5)2 2(x + 2)2 18(x – )2 4 3 5(x + 6)2 or 5(x – 6)2 27(x + )2 or 27(x – )2 2 3 36(x + )2 or 36(x – )2 7 3 Solution follows…

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**Completing the Square 7.2.1 Topic Independent Practice**

23. The length of a rectangle is twice its width. If the area can be found by completing the square for (18x2 + 60x) ft2, find the width of the rectangle. (3x + 5) ft Solution follows…

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**Completing the Square 7.2.1 Topic Round Up**

OK, so now you know how to add a number to a binomial to make a perfect square trinomial. In the next Topic you’ll learn how to convert any quadratic expression into perfect square trinomial form — and then in Topic you’ll use this to solve quadratic equations.

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