SOLVING QUADRATIC EQUATIONS

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
  3. Quadratic Formula
  4. Graphing

• All methods start with setting the equation equal to zero.

Solve for x in the following equation.

Problem 4.1b

Answer: x = - 8 , 4

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.

The answers are

Method 2: Completing the square

Add 32 to both sides of the equation.

Add 4 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,

Subtract 2 from both sides of the equation.

and

Method 3: Quadratic Formula

The quadratic formula is

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, +4 for b, and -32 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 4 and The answers are 4 and

Check these answers in the original equation.

Check the answer x=4 by substituting 4 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 4 for x, then x=4 is a solution.

Check the solution x=-8 by substituting -8 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 -8 for x, then x=-8 is a solution.

The solutions to the equation are 4 and -8.

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

If you would like to go back to the equation table of contents, click on Contents.

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