UNIT 10.5 SOLVING QUADRATICS UNIT 10.5 SOLVING QUADRATICS BY COMPLETING THE SQUAREBY COMPLETING THE SQUARE
Warm Up
Simplify.
19 1. 2.
3. 4.
Warm Up
Solve each quadratic equation by factoring.
5. x2 + 8x + 16 = 0
6. x2 – 22x + 121 = 0
7. x2 – 12x + 36 = 0
x = –4
x = 11
x = 6
Solve quadratic equations by completing the square.
Objective
completing the square
Vocabulary
In the previous lesson, you solved quadratic equations by isolating x2 and then using square roots. This method works if the quadratic equation, when written in standard form, is a perfect square.
When a trinomial is a perfect square, there is a relationship between the coefficient of the x-term and the constant term.
X2 + 6x + 9 x2 – 8x + 16 Divide the coefficient of the x-term by 2, then square the result to get the constant term.
An expression in the form x2 + bx is not a perfect square. However, you can use the relationship shown above to add a term to x2 + bx to form a trinomial that is a perfect square. This is called completing the square.
Example 1: Completing the Square
Complete the square to form a perfect square trinomial.
A. x2 + 2x + B. x2 – 6x +
x2 + 2x
x2 + 2x + 1
x2 + –6x
x2 – 6x + 9
Identify b.
.
Check It Out! Example 1
Complete the square to form a perfect square trinomial.
a. x2 + 12x + b. x2 – 5x +
x2 + 12x
x2 + 12x + 36
x2 + –5xIdentify b.
x2 – 6x +
.
Check It Out! Example 1
Complete the square to form a perfect square trinomial.
c. 8x + x2 +
x2 + 8x
x2 + 12x + 16
Identify b.
.
To solve a quadratic equation in the form x2 + bx = c, first complete the square of x2 + bx. Then you can solve using square roots.
Solving a Quadratic Equation by Completing the Square
Example 2A: Solving x2 +bx = c
Solve by completing the square.x2 + 16x = –15
Step 1 x2 + 16x = –15
Step 2
Step 3 x2 + 16x + 64 = –15 + 64
Step 4 (x + 8)2 = 49
Step 5 x + 8 = ± 7
Step 6 x + 8 = 7 or x + 8 = –7 x = –1 or x = –15
The equation is in the form x2 + bx = c.
Complete the square.
Factor and simplify.
Take the square root of both sides.
Write and solve two equations.
.
Example 2A Continued
Solve by completing the square.
x2 + 16x = –15
The solutions are –1 and –15.
Check x2 + 16x = –15
(–1)2 + 16(–1) –15
1 – 16 –15–15 –15
x2 + 16x = –15
(–15)2 + 16(–15) –15
225 – 240 –15–15 –15
Example 2B: Solving x2 +bx = c
Solve by completing the square.x2 – 4x – 6 = 0
Step 1 x2 + (–4x) = 6
Step 3 x2 – 4x + 4 = 6 + 4
Step 4 (x – 2)2 = 10
Step 5 x – 2 = ± √10
Write in the form x2 + bx = c.
Complete the square.
Factor and simplify.
Take the square root of both sides.
Write and solve two equations.
Step 6 x – 2 = √10 or x – 2 = –√10 x = 2 + √10 or x = 2 – √10
.Step 2
Example 2B Continued
Solve by completing the square.
The solutions are 2 + √10 and x = 2 – √10.
Check Use a graphing calculator to check your answer.
Check It Out! Example 2a
Solve by completing the square.
x2 + 10x = –9
Step 1 x2 + 10x = –9
Step 3 x2 + 10x + 25 = –9 + 25 Complete the square.
The equation is in the form x2 + bx = c.
Step 2
Step 4 (x + 5)2 = 16
Step 5 x + 5 = ± 4
Step 6 x + 5 = 4 or x + 5 = –4 x = –1 or x = –9
Factor and simplify.
Take the square root of both sides.
Write and solve two equations.
.
Check It Out! Example 2a Continued
Solve by completing the square.
x2 + 10x = –9 The solutions are –9 and –1.
x2 + 10x = –9
(–9)2 + 10(–9) –9
81 – 90 –9–9 –9
x2 + 16x = –15
(–1)2 + 16(–1) –15
1 – 16 –15–15 –15
Check
Check It Out! Example 2b
Solve by completing the square.
t2 – 8t – 5 = 0
Step 1 t2 + (–8t) = 5
Step 3 t2 – 8t + 16 = 5 + 16 Complete the square.
Write in the form x2 + bx = c.
Step 2
Step 4 (t – 4)2 = 21
Step 5 t – 4 = ± √21
Factor and simplify.
Take the square root of both sides.
Write and solve two equations.
Step 6 t = 4 + √21 or t = 4 – √21
.
Check It Out! Example 2b Continued
Solve by completing the square.
t = 4 – √21 or t = 4 + √21.The solutions are
Check Use a graphing calculator to check your answer.
Example 3A: Solving ax2 + bx = c by Completing the Square
Solve by completing the square.
–3x2 + 12x – 15 = 0
Step 1
x2 – 4x + 5 = 0x2 – 4x = –5
x2 + (–4x) = –5
Step 3 x2 – 4x + 4 = –5 + 4
Divide by – 3 to make a = 1.
Write in the form x2 + bx = c.
Complete the square.
.Step 2
Example 3A Continued
Solve by completing the square.
–3x2 + 12x – 15 = 0
Step 4 (x – 2)2 = –1
There is no real number whose square is negative, so there are no real solutions.
Factor and simplify.
Example 3B: Solving ax2 + bx = c by Completing the Square
Solve by completing the square.5x2 + 19x = 4
Step 1 Divide by 5 to make a = 1.
Write in the form x2 + bx = c.
Step 2 .
Complete the square.Step 3
Example 3B Continued
Solve by completing the square.
Factor and simplify.Step 4
Step 5Take the square root
of both sides.
Rewrite using like denominators.
Example 3B Continued
Solve by completing the square.
Step 6
The solutions are and –4.
Write and solve two equations.
Check It Out! Example 3a
Solve by completing the square.3x2 – 5x – 2 = 0
Step 1 Divide by 3 to make a = 1.
Write in the form x2 + bx = c.
Complete the square.
Factor and simplify.
Step 3
Step 4
Check It Out! Example 3a Continued
Solve by completing the square.
Step 2 .
Check It Out! Example 3a Continued
Solve by completing the square.
Write and solve two equations.
Step 6
−
Take the square root of both sides.
Step 5
Check It Out! Example 3b
Solve by completing the square.4t2 – 4t + 9 = 0
Step 1 Divide by 4 to make a = 1.
Write in the form x2 + bx = c.
Check It Out! Example 3b ContinuedSolve by completing the square.
4t2 – 4t + 9 = 0
Step 2
Step 3
Factor and simplify.
Complete the square.
Step 4
There is no real number whose square is negative, so there are no real solutions.
.
Example 4: Problem-Solving Application
A rectangular room has an area of 195 square feet. Its width is 2 feet shorter than its length. Find the dimensions of the room. Round to the nearest hundredth of a foot, if necessary.
11 Understand the Problem
The answer will be the length and width of the room.List the important information:
• The room area is 195 square feet.• The width is 2 feet less than the length.
22 Make a Plan
Example 4 Continued
Set the formula for the area of a rectangle equal to 195, the area of the room. Solve the equation.
Example 4 Continued
Solve33
Let x be the width.Then x + 2 is the length.
Use the formula for area of a rectangle.l • w = A
length times width = area of room
x + 2 • x = 195
Step 1 x2 + 2x = 195
Step 2
Step 3 x2 + 2x + 1 = 195 + 1
Step 4 (x + 1)2 = 196
Simplify.
Complete the square by adding 1 to both sides.
Factor the perfect-square trinomial.
Example 4 Continued
Take the square root of both sides.
Step 5 x + 1 = ± 14
Step 6 x + 1 = 14 or x + 1 = –14 Write and solve two equations.x = 13 or x = –15
.
Negative numbers are not reasonable for length, so x = 13 is the only solution that makes sense.
The width is 13 feet, and the length is 13 + 2, or 15, feet.
Example 4 Continued
Look Back44
The length of the room is 2 feet greater than the width. Also 13(15) = 195.
Check It Out! Example 4 An architect designs a rectangular room with an area of 400 ft2. The length is to be 8 ft longer than the width. Find the dimensions of the room. Round your answers to the nearest tenth of a foot.
11 Understand the Problem
The answer will be the length and width of the room.List the important information:
• The room area is 400 square feet.• The length is 8 feet more than the width.
22 Make a Plan
Set the formula for the area of a rectangle equal to 400, the area of the room. Solve the equation.
Check It Out! Example 4 Continued
Solve33
Let x be the width.Then x + 8 is the length.
Use the formula for area of a rectangle.
l • w = A
Check It Out! Example 4 Continued
length times width = area of room
X + 8 • x = 400
Step 1 x2 + 8x = 400
Step 3 x2 + 8x + 16 = 400 + 16
Step 4 (x + 4)2 = 416
Simplify.
Complete the square by adding 16 to both sides.
Factor the perfect-square trinomial.
Step 2
Check It Out! Example 4 Continued
Take the square root of both sides.
Step 5 x + 4 ≈ ± 20.4
Step 6 x + 4 ≈ 20.4 or x + 4 ≈ –20.4Write and solve two
equations.x ≈ 16.4 or x ≈ –24.4
.
Negative numbers are not reasonable for length, so x ≈ 16.4 is the only solution that makes sense.
The width is approximately16.4 feet, and the length is 16.4 + 8, or approximately 24.4, feet.
Check It Out! Example 4 Continued
Look Back44
The length of the room is 8 feet longer than the width. Also 16.4(24.4) = 400.16, which is approximately 400.
Complete the square to form a perfect square trinomial.
1. x2 +11x +
2. x2 – 18x +
Solve by completing the square.
3. x2 – 2x – 1 = 0
4. 3x2 + 6x = 144
5. 4x2 + 44x = 23
Lesson Quiz: Part I
81
6, –8
Lesson Quiz: Part II
6. Dymond is painting a rectangular banner for a football game. She has enough paint to cover 120 ft2. She wants the length of the banner to be 7 ft longer than the width. What dimensions should Dymond use for the banner?
8 feet by 15 feet
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