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

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§ 13.7

Quadratic Equations and Problem Solving

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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

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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

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Finding an Unknown Number

Example continued

2.) Translate

Continued

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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)

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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.

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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

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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.