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 4.3d:

Answer: exact answers;

and approximate answers

Solution:

The equation is already set to zero.

Rewrite the original equation as an equivalent equation without denominators by multiplying both sides by 16.

Simplify the left sides of the equation by combining like terms.

Method 1:Factoring

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

Method 2:Completing the square

Add 173 to both sides of the equation .

Divide both sides by 128:

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 128 for a , -204 for b, and -173 for c in the quadratic formula and simplify.

Method 4:Graphing

Graph (formed by subtracting the right side of the original equation from the left side of the original equation). Graph y=0 (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 two x-intercepts located at 2.20633314611 and -0.612583146106. This means that there are two real answers: x=2.20633314611 and

The answers are 2.20633314611 and -0.612583146106. 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=2.20633314611 by substituting 2.20633314611 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 2.20633314611 for x, then x=2.20633314611 is a solution.

Check the solution x=-0.612583146106 by substituting -0.612583146106 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 -0.612583146106 for x, then x=-0.612583146106 is a solution.

The solutions to the equation are

Comment:You can use the exact solutions to factor the left side of the

For example, since , then

Since , then

Since the product

and

However not the only factors:

Since the first term of the product is not there must be another factor of 8:

Let s check to see whether

Therefore 8x - x- is factored as 8

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