Section P7Equations

Solving Linear Equations in

One Variable

Example

Solve the equation: 2(3x-5)=-5(x-1)+x

Linear Equations with Fractions

Solving with Fractions

Example

1 3 2 1Solve for x.

3 2 12

x x

Example

3 1 2Solve for x.

5 2 5

x x x

Rational Equations

A rational equation is an equation containing one

or more rational expressions. Notice how the variables

appear in the denominator in rational equations and the

previous examples ( Linear Equations with Fractions)

only had variables in the numerator.

Solving Rational Equations

Example

2

3 4 1Solve the Rational Equation.

2 24x xx

Example

5 2 1Solve the Rational Equation.

3 2x x

Example

2

3 2Solve the Rational Equation. 6

1 1x x

Solving a Formula for One of Its Variables

A-Prt=P

A=Prt+P

A=P(rt+1)

AP or

rt+1A

P=rt+1

Example

Solve for in the formula for the Perimeter of a rectangle.l

Example1

Solve for in the formula for the area of a triangle A= bh.2

b

Equations Involving

Absolute Value

Example

Solve: 5 9 7x

Example

Solve: 2 2 7 14 0x

Quadratic Equations and Factoring

Example

Solve the equation (2x-5)(3x+4)=0 using the Zero-Product

Principle.

Example

2Solve the equation by factoring: x 3 4 0x

Example

2Solve the equation by factoring: 2x 7 4 0x

Quadratic Equations and the Square Root Property

Example

2

Solve the following problem by the square root property.

(x-4) 25

Example

2

Solve the following problem by the square root property.

4x 7 0

Quadratic Equations and Completing the Square

Start Add Result Factored Form

222 2

222 2

222 2

1x 6 6 9 x 6 9 3

2

1x 4 4 4 x 4 4 2

2

1x 20 20 100 x 20 100 10

2

x x x

x x x

x x x

g

g

g

Obtaining a Perfect Square Trinomial

21

2b

Completing the Square

Example

2

Complete the square to solve the following problem.

x 10 3 0x

Example

2

Complete the square to solve the following problem.

x 8 13 0x

Example

2

Complete the square to solve the following problem.

x 5 10x

Quadratic Equations and the Quadratic Formula

Example

2

Solve the equation using the quadratic formula.

x 6 3x

Example

2

Solve the equation using the quadratic formula.

2x 4 5x

Quadratic Equations and

the Discriminant

Example

2

2

2

Use the discriminant to find the number and types

of solutions, but don't solve the equation.

a. x 5 6 0

b. x 3 9

c. 2x 4 9

x

x

x

Graphing CalculatorThe real solutions of a quadratic equation ax2+bx+c=0 correspond to the x-intercepts of the graph. The U shaped graph shown below has two x intercepts. When y=0, the value(s) of x will be the solution to the equation. Since y=0 these are called the zeros of the function.

Solving Polynomial Equations using the Graphing Calculator

Repeat this process for each x intercept.

By pressing 2nd Trace to get Calc, then the #2,you get the zeros. It will ask you for left and right bounds, and then a guess. For left and right bounds move the blinking cursor (using the arrow keys-cursor keys) to the left and press enter. Then move the cursor to the right of the x intercept and press enter. Press enter when asked to guess. Then you get the zeros or solution.

Determining Which

Method to Use

Example

2Factor and solve. -3x 6 0x

Example

2Solve by any method. -3x 15 0

Example

2Solve by any method. x 4 10 0x

Radical Equations

A radical equation is an equation in which the variable occurs in a square root, cube root, or any higher root. We solve the equation by squaring both sides.

This new equation has two solutions, -4 and 4. By contrast, only 4 is a solution of the original equation, x=4. For this reason, when raising both sides of an equation to an even power, check proposed solutions in the original equation.

Extra solutions may be introduced when you raise both sides of a radical equation to an even power. Such solutions, which are not solutions of the given equation are called extraneous solutions or extraneous roots.

2

4

If we square both sides, we obtain

x 16

16 -4 or 4

x

x

Example

Solve and check your answers:

5 1x x

(a)

(b)

(c)

(d)

1Solve for h in the area formula for a trapezoid. A= ( )

2h a b

2

2( )

2

A

a bA

a b

A

a bA

a b

(a)

(b)

(c)

(d)

Solve: 3 8 27 0x

3, 3

4,10

1, 17

17, 1

x

x

x

x