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.

Example 2:

Set the equation equal to zero by subtracting 3x and 10 from both sides of the equation.

Method 1:Factoring

Method 2:Completing the square

Add 7 to both sides of the equation

Divide both sides by 12 :

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 :

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.Substitute 12 for a, -25 for b , and -7 for c in the quadratic formula and simplify.

Method 4: Graphing

Graph (This equation is formed by subtracting the right side of the original graph from the left side of the original graph.) Graph The graph of 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.

You can see from the graph that there are two x-intercepts located at 2.333333 and -0.25. This means that there are two real answers: x=2.333333 and -0.25.

The answers are 2.333333 and -0.25. These answers may or may not be solutions to the original equation. You must check the answers with the original equation.

Check these answers in the original equation.

Check the solution x=2.333333 by substituting 2.333333 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 x = 2.333333 for x, then x = 2.333333 is a solution.

Check the solution x=-0.25 by substituting -0.25 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 -0.25 for x, then x=-0.25 is a solution.

The solutions to the equation are - 0.25 and 2.333333.

If you would like to work another example, click on Example.

If you would like to test yourself by working some problems similar to this example, click on Problem.

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

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