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 4.1D

Answer:

Solution: The equation is already equal to zero.

Method 1: Factoring

The equation cannot be easily factored.

Method 2: Completing the square

Subtract 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 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 51 for a, for b, and 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 -0.398858303987 and The answers are -0.398858303987 and

Check these answers in the original equation.

Check the answer x=-0.398858303987 by substituting -0.398858303987 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 -0.398858303987 for x, then x=-0.398858303987 is a solution.

Check the solution x=-0.156697251568 by substituting -0.156697251568 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 -0.156697251568 for x, then x=-0.156697251568 is a solution.

The solutions to the equation are and

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

are factors of

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