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Factoring Polynomials
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We set each factor equal to 0 and solve for x.

4x + 3 = 0 or x + 2 = 0

4x = 3 or x = 2

x = ¾ or x = 2

So the x-intercepts are the points (¾, 0) and (2, 0).

Finding x-intercepts

Example

Slide 58

 13.7

13.7

Quadratic Equations and Problem Solving

Slide 59

Strategy for Problem Solving

Strategy for Problem Solving

General Strategy for Problem Solving

Understand the problem

Read and reread the problem

Choose a variable to represent the unknown

Construct a drawing, whenever possible

Propose a solution and check

Translate the problem into an equation

Solve the equation

Interpret the result

Check proposed solution in problem

State your conclusion

Slide 60

Finding an Unknown Number

Finding an Unknown Number

The product of two consecutive positive integers is 132. Find the two integers.

1.) Understand

Read and reread the problem. If we let

x = one of the unknown positive integers, then

x + 1 = the next consecutive positive integer.

Example

Continued

Slide 61

Finding an Unknown Number

Finding an Unknown Number

Example continued

2.) Translate

Continued

Slide 62

Finding an Unknown Number

Finding an Unknown Number

Example continued

3.) Solve

Continued

x(x + 1) = 132

x2 + x = 132 (Distributive property)

x2 + x 132 = 0 (Write quadratic in standard form)

(x + 12)(x 11) = 0 (Factor quadratic polynomial)

x + 12 = 0 or x 11 = 0 (Set factors equal to 0)

x = 12 or x = 11 (Solve each factor for x)

Slide 63

Finding an Unknown Number

Finding an Unknown Number

Example continued

4.) Interpret

Check: Remember that x is suppose to represent a positive integer. So, although x = -12 satisfies our equation, it cannot be a solution for the problem we were presented.

If we let x = 11, then x + 1 = 12. The product of the two numbers is 11 12 = 132, our desired result.

State: The two positive integers are 11 and 12.

Slide 64

Pythagorean Theorem

Pythagorean Theorem

In a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse.

(leg a)2 + (leg b)2 = (hypotenuse)2

The Pythagorean Theorem

Slide 65

Find the length of the shorter leg of a right triangle if the longer leg is 10 miles more than the shorter leg and the hypotenuse is 10 miles less than twice the shorter leg.

Find the length of the shorter leg of a right triangle if the longer leg is 10 miles more than the shorter leg and the hypotenuse is 10 miles less than twice the shorter leg.

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