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 4.4c:

Answer:

Solution:

The equation is already equal to zero.

If you have forgotten how to manipulate fractions, click on Fractions for a review.

Remove all the fractions by writing the equation in an equivalent form without fractional coefficients. In this problem, you can do it by multiplying both sides of the equation by 8.

Method 1:Factoring

The equation is not easily factored. Therefore, we will not use this method.

Method 2:Completing the square

Subtract 192 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:

Subtract 7 from both sides of the equation:

The answers are 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. Substitute for a, for b, and for c in the quadratic formula and simplify.

The answers are and

Method 4:Graphing

Graph the equation, (the left side of the original equation). Graph (the right side of the original equation and the x-axis). 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 no x-intercepts. This means that there are no real answers; the answers are imaginary.

The answers are These answers may or may not be solutions to the original equations. You must verify that these answers are solutions.

Check these answers in the original equation.

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

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

The solutions to the equation are

Comment:You can use the exact solutions to factor the left side of the original equation minus the right side of the original equation:

Since

Since

The product

Since and

then we could say

However the product of the first terms of the factors does not equal

Multiply by

Let s check to see if

The factors of are , and

If you would like to review the solution to problem 4.4d, click on Problem

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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