algebra 1 lesson 9-6 warm-up. algebra 1 “completing the square” (9-6) what does it mean to...

ALGEBRA 1

Lesson 9-6 Warm-Up

ALGEBRA 1

“Completing the Square” (9-6)

What does it mean to “complete the square”?

Review: So far, we discussed two methods to solving quadratic equations in the form of ax2

+ bx + c = 0. However, they can only be used in some cases.

1. Finding square roots (9-4) - Can only be used when b = 0.

2. factoring (9-5) – can only be used when you can FOIL the equation

Completing the Square: This is a third method for solving quadratic equations that that turns ax2 + bx + c = 0 into m2 = n. By turning the left side of the equation into a perfect square, you can solve these equations by finding their square roots. This works for every quadratic equation. Below is a model of how completing the square works.

ALGEBRA 1


How do you “complete the square” when the equation is in the form of x2 + bx = c?

Algebraically, the model is written:

x2 + bx + c = 0

x2 + 8x = 0 There is no c (c = 0)

x2 + 8x + 16 = 0 + 16 Add 16 to both sides to complete the square

(x + 4)2 = 16 Factor

m2 = n Now, you can solve for x by taking the square root of both sides.

Rule: Notice that the square root of the c term,16, is 4, which is half of the b term, 8. Therefore, to find the c term that will complete the square, take half of the b term and square it:

c =

Note: When you complete the square, you must add the “c” term to both sides of the equation using the Addition Property of Equality.

Example: Write the left side of x2 + 7x = 62 as a perfect square.

x2 + 7x + = 62 +

b2

2

2 72

272

ALGEBRA 1

Find the value of c to complete the square for x2 – 16x + c.

The value of b in the expression x2 – 16x + c is –16.

The term to add to x2 – 16x is or 64. –16

22

Completing the SquareLESSON 9-6

Additional Examples

ALGEBRA 1

Solve the equation x2 + 5x = 50.

Step 1: Write the left side of x2 + 5x = 50 as a perfect square.

x2 + 5x = 50

x2 + 5x + = 50 +52

2 52

2 Add , or , to each side of the

equation.

52

2 254

52

2x + 200

4 +254= Write x2 + 5x + as a square. 5

2

2

Rewrite 50 as a fraction with denominator 4.

52

2x + =

2254


Additional Examples

ALGEBRA 1

(continued)

Step 2: Solve the equation.

Find the square root of each side.52

x + = 2254

±

52

x + = 152

± Simplify.

52

x + = 152 or

52x + =

152– Write as two equations.

x = 5 or x = –10 Solve for x.


Additional Examples

ALGEBRA 1


How do you “complete the square” when the equation is in the form of x2 + bx + c = 0?

If the equation is in the form of x2 + bx + c = 0 and you want to use the “complete the square” method, move the “c” term to the other side of the equal sign using the Addition or Subtraction Property of Equality. Then, complete the square.

Example: Solve x2 + 9x = 136.

ALGEBRA 1

Solve x2 + 10x – 16 = 0 . Round to the nearest hundredth.

Step 1: Rewrite the equation in the form x2 + bx = c and complete the square.

x2 + 10x – 16 = 0

x2 + 10x = 16 Add 16 to each side of the equation.

(x + 5)2 = 41 Write x2 + 10x +25 as a square.

x2 + 10x + 25 = 16 + 25 Add , or 25, to each side of the equation.102

2


Additional Examples

ALGEBRA 1

(continued)

Step 2: Solve the equation.

x + 5 = ± 41 Find the square root of each side.

Use a calculator to find 41x + 5 ± 6.40

x + 5 6.40 or x + 5 –6.40 Write as two equations.

Subtract 5 from each side.x 6.40 – 5 or x –6.40 – 5

x 1.40 or x –11.40 Simplify


Additional Examples

ALGEBRA 1

Suppose you wish to section off a soccer field as shown

in the diagram to run a variety of practice drills. If the area of the

field is 6000 yd2, what is the value of x?

Define: width = x + 10 + 10 = x + 20length = x + x + 10 + 10 = 2x + 20

Words: length width = area

Equation: (2x + 20)(x + 20) = 6000 2x2 + 60x + 400 = 6000

Step 1: Rewrite the equation in the form x2 + bx = c.

2x2 + 60x + 400 = 6000

2x2 + 60x = 5600 Subtract 400 from each side.

x2 + 30x = 2800 Divide each side by 2.


Additional Examples

ALGEBRA 1

(continued)

Step 2: Complete the square.

x2 + 30x + 255 = 2800 + 225 Add , or 225, to each side.302

2

(x + 15)2 = 3025 Write x2 + 30x + 255 as a square.


Additional Examples

Step 3: Solve each equation.

The value of x is 40 yd.

(x + 15) = ± 3025 Take the square root of each side.

x + 15 = ± 55

x + 15 = 55 or x + 15 = –55

x = 40 or x = –70 Use the positive answer for this problem.

Write as two equations.

ALGEBRA 1

1. x2 + 14x = –43

2. 3x2 + 6x – 24 = 0

3. 4x2 + 16x + 8 = 40

Solve each equation by completing the square. If necessary, round tothe nearest hundredth.

–9.45, –4.55

–4, 2

–5.46, 1.46


Lesson Quiz

algebra 1 lesson 9-6 warm-up. algebra 1 “completing the square” (9-6) what does it mean to...

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