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 4.2d:

Answer:

approximate solutions

Solution:

Remove the denominators from the original equation by multiplying both sides by 144.

Set the equation equal to zero by subtracting 2016x and 1584 from both sides of the equation.

Method 1:Factoring

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

Method 2:Completing the square

Subtract 18 from both sides of the equation .

Add 1575 to both sides of the equation :

Divide both sides by 144:

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 from 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. Simply insert 144 for a, -1936 for b, and -1575 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 14.2139358 and -0.769491307. This means that there are two real answers: x = 14.2139358 and -0.769491307.

The answers are x = 14.2139358 and - 0.769491307. 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=14.2139358 by substituting 14.2139358 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 14.2139358 for x, then x=14.2139358 is a solution.

Check the solution x=-0.769491307 by substituting -0.769491307 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 -0.769491307 for x, then x=-0.769491307 is a solution.

The solutions to the equation are 14.2139358 and

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