Introduction A trinomial of the form that can be written as the square of a binomial is called a...
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Transcript of Introduction A trinomial of the form that can be written as the square of a binomial is called a...
Introduction
A trinomial of the form that can be written
as the square of a binomial is called a perfect square
trinomial. We can solve quadratic equations by
transforming the left side of the equation into a perfect
square trinomial and using square roots to solve.
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5.2.3: Completing the Square
Key Concepts• When the binomial (x + a) is squared, the resulting
perfect square trinomial is x2 + 2ax + a2. • When the binomial (ax + b) is squared, the resulting
perfect square trinomial is a2x2 + 2abx + b2.
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5.2.3: Completing the Square
Key Concepts, continued
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5.2.3: Completing the Square
Completing the Square to Solve Quadratics
1. Make sure the equation is in standard form, ax2 + bx + c.
2. Subtract c from both sides. 3. Divide each term by a to get a leading
coefficient of 1. 4. Add the square of half of the coefficient of the
x-term to both sides to complete the square. 5. Express the perfect square trinomial as the
square of a binomial. 6. Solve by using square roots.
Common Errors/Misconceptions• neglecting to add the c term of the perfect square
trinomial to both sides
• not isolating x after squaring both sides
• forgetting that, when taking the square root, both the positive and negative roots must be considered (±)
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5.2.3: Completing the Square
Guided Practice
Example 2Solve x2 + 6x + 4 = 0 by completing the square.
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5.2.3: Completing the Square
Guided Practice: Example 2, continued
1. Determine if x2 + 6x + 4 is a perfect square trinomial. Take half of the value of b and then square the result. If this is equal to the value of c, then the expression is a perfect square trinomial.
x2 + 6x + 4 is not a perfect square trinomial because the square of half of 6 is not 4.
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5.2.3: Completing the Square
Guided Practice: Example 2, continued
2. Complete the square.
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5.2.3: Completing the Square
x2 + 6x + 4 = 0 Original equation
x2 + 6x = –4 Subtract 4 from both sides.
x2 + 6x + 32 = –4 + 32
Add the square of half of the coefficient of the x-term to both sides to complete the square.
x2 + 6x + 9 = 5 Simplify.
Guided Practice: Example 2, continued
3. Express the perfect square trinomial as the square of a binomial. Half of b is 3, so the left side of the equation can be written as (x + 3)2.
(x + 3)2 = 5
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5.2.3: Completing the Square
Guided Practice: Example 2, continued
4. Isolate x.
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5.2.3: Completing the Square
(x + 3)2 = 5 Equation
Take the square root of both sides.
Subtract 3 from both sides.
Guided Practice: Example 2, continued
5. Determine the solution(s). The equation x2 + 6x + 4 = 0 has two solutions,
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5.2.3: Completing the Square
✔
Guided Practice
Example 3Solve 5x2 – 50x – 120 = 0 by completing the square.
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5.2.3: Completing the Square
Guided Practice: Example 3, continued
1. Determine if 5x2 – 50x – 120 = 0 is a perfect square trinomial. The leading coefficient is not 1.
First divide both sides of the equation by 5 so that a = 1.
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5.2.3: Completing the Square
5x2 – 50x – 120 = 0 Original equation
x2 – 10x – 24 = 0 Divide both sides by 5.
Guided Practice: Example 3, continuedNow that the leading coefficient is 1, take half of the value of b and then square the result. If the expression is equal to the value of c, then it is a perfect square trinomial.
5x2 – 50x – 120 = 0 is not a perfect square trinomial because the square of half of –10 is not –24.
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5.2.3: Completing the Square
Guided Practice: Example 3, continued
2. Complete the square.
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5.2.3: Completing the Square
x2 – 10x – 24 = 0 Equation
x2 – 10x = 24 Add 24 to both sides.
x2 – 10x + (–5)2 = 24 + (–5)2
Add the square of half of the coefficient of the x-term to both sides to complete the square.
x2 – 10x + 25 = 49 Simplify.
Guided Practice: Example 3, continued
3. Express the perfect square trinomial as the square of a binomial. Half of b is –5, so the left side of the equation can be written as (x – 5)2.
(x – 5)2 = 49
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5.2.3: Completing the Square
Guided Practice: Example 3, continued
4. Isolate x.
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5.2.3: Completing the Square
(x – 5)2 = 49 Equation
Take the square root of both sides.
Add 5 to both sides.
x = 5 + 7 = 12 orx = 5 – 7 = –2
Split the answer into two separate equations and solve for x.
Guided Practice: Example 3, continued
5. Determine the solution(s). The equation 5x2 – 50x – 120 = 0 has two solutions, x = –2 or x = 12.
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5.2.3: Completing the Square
✔