Solve quadratic equations

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Solve quadratic equations EXAMPLE 5 Solve quadratic equations Solve (a) 3x2 + 10x – 8 = 0 and (b) 5p2 – 16p + 15 = 4p – 5. a. 3x2 + 10x – 8 = 0 Write original equation. (3x – 2) (x + 4) = 0 Factor. 3x – 2 = 0 or x + 4 = 0 Zero product property or x = – 4 x = 23 Solve for x.

Solve quadratic equations EXAMPLE 5 Solve quadratic equations (b) 5p2 – 16p + 15 = 4p – 5. b. 5p2 – 16p + 15 = 4p – 5. Write original equation. 5p2 – 20p + 20 = 0 Write in standard form. p2 – 4p + 4 = 0 Divide each side by 5. (p – 2)2 = 0 Factor. p – 2 = 0 Zero product property p = 2 Solve for p.

EXAMPLE 6 Use a quadratic equation as a model Quilts You have made a rectangular quilt that is 5 feet by 4 feet. You want to use the remaining 10 square feet of fabric to add a decorative border of uniform width to the quilt. What should the width of the quilt’s border be?

Use a quadratic equation as a model EXAMPLE 6 Use a quadratic equation as a model SOLUTION Write a verbal model. Then write an equation. 10 = 20 + 18x + 4x2 – 20 Multiply using FOIL. 0 = 4x2 + 18x – 10 Write in standard form 0 = 2x2 + 9x – 5 Divide each side by 2. 0 = (2x – 1) (x + 5) Factor. 2x – 1 = 0 or x + 5 = 0 Zero product property 12 x = or x = – 5 Solve for x.

EXAMPLE 6 Use a quadratic equation as a model ANSWER Reject the negative value, – 5. The border’s width should be ft, or 6 in. 12

EXAMPLE 7 Solve a multi-step problem Magazines A monthly teen magazine has 28,000 subscribers when it charges $10 per annual subscription. For each $1 increase in price, the magazine loses about 2000 subscribers. How much should the magazine charge to maximize annual revenue ? What is the maximum annual revenue ?

EXAMPLE 7 Solve a multi-step problem SOLUTION STEP 1 Define the variables. Let x represent the price increase and R(x) represent the annual revenue. STEP 2 Write a verbal model. Then write and simplify a quadratic function.

EXAMPLE 7 Solve a multi-step problem R(x) = (– 2000x + 28,000) (x + 10) R(x) = – 2000(x – 14) (x + 10)

EXAMPLE 7 Solve a multi-step problem STEP 3 Identify the zeros and find their average. Find how much each subscription should cost to maximize annual revenue. The zeros of the revenue function are 14 and –10. The average of the zeroes is 14 + (– 1 0) 2 = 2. To maximize revenue, each subscription should cost $10 + $2 = $12. STEP 4 Find the maximum annual revenue. R(2) = – 2000(2 – 14) (2 + 10) = $288,000

EXAMPLE 7 Solve a multi-step problem ANSWER The magazine should charge $12 per subscription to maximize annual revenue. The maximum annual revenue is $288,000.

GUIDED PRACTICE GUIDED PRACTICE for Examples 5, 6 and 7 Solve the equation. 19. 6x2 – 3x – 63 = 0 6x2 – 3x – 63 = 0 Write original equation. 2x2 – x – 21 = 0 Divide each side by 3. (2x – 7) (x + 3) = 0 Factor. 2x – 7 = 0 or x + 3 = 0 Zero product property or x = – 3 x = 72 = 3 12 Solve for x.

GUIDED PRACTICE GUIDED PRACTICE for Examples 5, 6 and 7 20. 12x2 + 7x + 2 = x +8 12x2 + 7x + 2 = x +8 Write original equation. 12x2 + 6x – 6 = 0 Write in standard form 4x2 + 2x – 2 = 0 Divide each side by 3 (2x + 2) (2x – 1) = 0 Factor. 2x + 2 = 0 or 2x – 1 = 0 Zero product property or x = 12 x = – 1 Solve for x.

GUIDED PRACTICE GUIDED PRACTICE for Examples 5, 6 and 7 21. 7x2 + 70x + 175 = 0 7x2 + 70x + 175= 0 Write original equation. 7x2 + 70x + 175 = 0 Write in standard form x2 + 10x + 25 = 0 Divide each side by 7 (x + 5) (x – 5) = 0 Factor. x + 5 = 0 or x – 5= 0 Zero product property or x = 5 x = – 5 Solve for x.

GUIDED PRACTICE GUIDED PRACTICE for Examples 5, 6 and 7 22. What If ? In Example 7, suppose the magazine initially charges $11 per annual subscription. How much should the magazine charge to maximize annual revenue ? What is the maximum annual revenue ? SOLUTION STEP 1 Define the variables. Let x represent the price increase and R(x) represent the annual revenue. STEP 2 Write a verbal model. Then write and simplify a quadratic function.

GUIDED PRACTICE GUIDED PRACTICE for Examples 5, 6 and 7 Annual Revenue = Annual Revenue Subscription price R(x) = (28,000 – 2000x) (11 + x) R(x) = (– 2000x + 28000) (x + 11) R(x) = – 2000 (x – 14) (x + 11) STEP 3 Identify the zeros and find their average. Find how much each subscription should cost to maximize annual revenue. The zeros of the revenue function are 14 and –11. The average of the zeroes is 14 + (– 11) 2 3 2. =

GUIDED PRACTICE GUIDED PRACTICE for Examples 5, 6 and 7 To maximize revenue, each subscription should cost $11 + = $12.50. 3 2. STEP 4 Find the maximum annual revenue. = $312,500 R( ) 3 2. = – 2000(2 – 14) ( + 11) ANSWER The magazine should charge $12.50 per subscription to maximize annual revenue. The maximum annual revenue is $312,500.