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:

The equation is already equal to zero.

Method 1: Factoring

The left side of the equation is not easily factored, so we will not use this method.

Method 2: Completing the square

Subtract 3 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.

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, -5 for b, and 3 for c in the quadratic formula and simplify

.

and

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 4.30277563773 and

The exact answers are and The approximate answers are 4.30277563773 and

Check these answers in the original equation.

Check the solution x=4.30277563773 by substituting 4.30277563773 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 4.30277563773 for x, then x=4.30277563773 is a solution.

Check the solution x=0.697224362268 by substituting 0.697224362268 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 0.697224362268 for x, then x=0.697224362268 is a solution.

The solutions to the equation are and rounded to 4.30277563773 and 0.697224362268

Comment: You can use the solutions to factor the original equation.

For example, since , then , and

Since , then , and

Since the product and , then we can say that This means that and
are factors of

If you would like to go 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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