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Solving Equations

Solving Linear Equations

A linear equation has the form ax + b = c, where a, and b are constants and x is the variable. To solve a linear equation, we must find the value of the variable which make the equation true. An algebraic method that can be used to solve linear equations is the following: Replace the equation with a simpler equivalent one and continue this process until a solution is reached. Below are some procedures that result in equivalent equations.

Interchange the two sides of the equation.

Simplify either or both sides of the equation by removing parenthesis, adding like terms, etc.

Add or subtract the same expression from both sides of the equation.

Multiply or divide both sides of the equation by the same non-zero expression.

 

The following examples illustrate these ideas.

Example

Solve the linear equation 100 = -7.2t + 250 for t. Round the answer to two decimal places if necessary.

Solution

Example

Solve the linear equation 3.2(0.7x + 12) +10.5 = -0.8x + 1.6. Round the answer to two decimal places.

Solution

  

Solving Quadratic Equations

Factoring Method

A quadratic equation has the form

If the expression  can be factored, then we can set each factor equal to zero. The following example illustrates the factoring method.

Example

Solve the quadratic equation

.

Solution


 

The Quadratic Formula

Since most quadratic expressions cannot be factored, the factoring method is limited in its usefulness. However, any quadratic equation that has real solutions can be solved by the quadratic formula

                                                      

The quantity  is called the discriminant of the quadratic equation because it determines whether the equation has one, two, or no real solutions:

If , the equation has two real solutions.

If , the equation has exactly one solution.

If , the equation has no real solutions.

Example

Use the quadratic formula to find the solutions, if any, of the quadratic equation

Solution

Comparing our equation to , we see that  Substituting these values into the quadratic formula we get

                                          

 Evaluating this expression, we find two solutions, x = -5.49 and x =1.17 (rounded to two decimal places).

Example                                                       

Use the quadratic formula to find the solutions, if any, of the equation

Solution

In this case, we must first subtract 160 from both sides so that the equation is in the form .

We now see that  Substituting these values into the quadratic formula, we get,

                                                .

Evaluating this expression yields the two solutions t= -0.71 and t = 3.52 (rounded to two decimal places).

Solving Equations Graphically

Almost any equation in one variable can be solved graphically. The strategy is as follows: graph the left hand side and the right hand side of the equation on the same coordinate axes, and then trace the graph to find the point(s) where the two curves intersect. If there is no point of intersection then we say that the equation has no solution. The following example will illustrate this procedure.

Example

Find all solutions of the equation . Round your answers to three decimal places.

Solution

First we will use the Plot-Solve tool to graph the left-hand side of the equation, and the right-hand side,  on the same coordinate axes. See Figure 1.

                                    Figure 1

The graph shows that there are two points of intersection. Click on one of the points and zoom in to find the coordinates of that point of intersection. Figure 2 shows that the intersection point on the left has coordinates (-3.3, 60).

                                    Figure 2

We can repeat the process to find the right intersection point. The coordinates are

(45.2, 60). See Figure 3.

                                    Figure 3

The equation has two solutions,  (accurate to one decimal place).