Note:

• A quadratic equation is a polynomial equation of degree 2.

• The ''U'' shaped graph of a quadratic is called a parabola.

• A quadratic equation has two solutions. Either two distinct real solutions, one double real solution or two imaginary solutions.

• There are several methods you can use to solve a quadratic equation:

1. Factoring
2. Completing the Square
4. Graphing

Solve for x in the following equation.

Problem 4.1a

Solution: The equation is already equal to zero.

Method 1: Factoring

The equation can be written in the equivalent form of The only way that a product can equal zero is if one or both of the factors equal zero.

Method 2: Completing the square

Subtract 10 from both sides of the equation.

Add to both sides of the equation.

Factor the left side and simplify the right side.

Take the square root of both sides of the equation,

Add to both sides of the equation.

In the equation , a is the coefficient of the term, b is the coefficient of the x term, and c is the constant. Simply insert 1 for a, -7 for b, and 10 for c in the quadratic formula and simplify.

Method 4: Graphing

Graph y= the left side of the equation or and graph y= the right side of the equation or y=0. The graph of y=0 is nothing more than the x-axis. So what you will be looking for is where the graph of crosses the x-axis. Another way of saying this is that the x-intercepts are the solutions to this equation. The x-intercepts are 2 and The answers are 2 and

Check these answers in the original equation.

Check the answer x=2 by substituting 2 in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

• Left Side:

• Right Side:

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 2 for x, then x=2 is a solution.

Check the solution x=5 by substituting 5 in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

• Left Side:

• Right Side:

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 5 for x, then x=5 is a solution.

The solutions to the equation are 2 and 5.

If you would like to review the solution to problem 4.1b, click on Problem