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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). 3.7 Complete the Square. 2 Reasons to Complete the Square. To solve quadratics To write the function from standard to vertex form. - PowerPoint PPT Presentation
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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)
3.7 Complete the Square
2 Reasons to Complete the Square
1. To solve quadratics
2. To write the function from standard to vertex form.
EXAMPLE 1 Solve a quadratic equation by finding square roots
Solve x2 – 8x + 16 = 25.
x2 – 8x + 16 = 25 Write original equation.
(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.
The solutions are 4 + 5 = 9 and 4 –5 = – 1.
ANSWER
GUIDED PRACTICE
1. x2 + 6x + 9 = 36.
3 and –9.ANSWER
Solve the equation by finding square roots.
2. x2 – 10x + 25 = 1.
4 and 6.ANSWER
3. x2 – 24x + 144 = 100.
2 and 22.ANSWER
GUIDED PRACTICE for Examples 1 and 2
Find the value of c that makes the expression a perfect square trinomial.Then write the expression as the square of a binomial.
4.
x2 + 14x + c
49 ; (x + 7)2ANSWER
5.
x2 + 22x + c
121 ; (x + 11)2ANSWER
6.
x2 – 9x + c
ANSWER ; (x – )2.814
92
To solve quadratics by completing the square. . .1. Write one side of the equation in the
form x2 + bx (move the c over)
2. Find the term to complete the square and add to both sides
3. When you add (b/2)2, you now can factor it into
4. Then, take the square root to solve.
2
2
b
2
2
bx
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 Add –122
2( ) = (–6)2= 36 to each side.
(x – 6)2 = 32 Write left side as a binomial squared.
Solve for x.
Take square roots of each side.x – 6 = + 32
x = 6 + 32
x = 6 + 4 2 Simplify: 32 = 16 2 = 4 2
The solutions are 6 + 4 and 6 – 42 2ANSWER
EXAMPLE 4 Solve ax2 + bx + c = 0 when a = 1
Solve 2x2 + 8x + 14 = 0 by completing the square.
2x2 + 8x + 14 = 0 Write original equation.
x2 + 4x + 7 = 0
Write left side in the form x2 + bx.
x2 + 4x + 4 = –7 + 4 Add 42
2( ) = 22 = 4 to each side.
(x + 2)2 = –3 Write left side as a binomial squared.
Solve for x.
Take square roots of each side.x + 2 = + –3
x = –2 + –3
x = –2 + i 3
x2 + 4x = –7
Divide each side by the coefficient of x2.
Write in terms of the imaginary unit i.
The solutions are –2 + i 3 and –2 – i 3 .
GUIDED PRACTICE
x2 + 6x + 4 = 0
–3+ 5ANSWER
7.
Solve the equation by completing the square.
x2 – 10x + 8 = 0
5 + 17ANSWER
8.
2n2 – 4n – 14 = 0
1 + 2 2ANSWER
9.
3x2 + 12x – 18 = 010.
–2 + 10ANSWER
11. 6x(x + 8) = 12
–4 +3 2ANSWER
1 + 26ANSWER
12. 4p(p – 2) = 100
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 + ? ) + 22Prepare to complete the square.
y + 25 = (x2 – 10x + 25) + 22Add –102
2( ) = (–5)2= 25 to each side.
y + 25 = (x – 5)2 + 22 Write x2 – 10x + 25 as a binomial squared.
y = (x – 5)2 – 3 Solve for y.
The vertex form of the function is y = (x – 5)2 – 3. The vertex is (5, –3).
ANSWER
GUIDED PRACTICE
y = x2 – 8x + 17
y = (x – 4)2 + 1 ; (4, 1).ANSWER
13.
Write the quadratic function in vertex form. Then identify the vertex.
y = x2 + 6x + 3
y = (x + 3)2 – 6 ; (–3, –6)
ANSWER
14.
f(x) = x2 – 4x – 4
y = (x – 2)2 – 8 ; (2 , –8)ANSWER
15.