Introduction

Quadratic Equation

Many times completing the square can begin to have a lot of fractions in the work. There is another method that can be easier when the a is not equal to 1.

A quadratic equation in standard from is given by:

  • ax2 + bx + c = 0.

When we solve an equation, we get x by itself. If we get x by itself from this equation, we could plug into that equation every time.

We are going to solve for x by completing the square.

Steps

Step 1: Subtract c from both sides.

  • ax2 + bx = −c.

Step 2: Divide everything by a.

  • a x squared over a plus b x over a equals negative c over a
  • x squared plus b over a x equals negative c over a

Step 3: Take half of b (this is the term in front of x) and square it.

  • one-half times b over a equals open paren b over 2 a close paren squared equals the fraction with numerator b squared and denominator 4 a squared

Step 4: Add this to both sides.

  • x squared plus b x over a plus the fraction with numerator b squared and denominator 4 a. equals the fraction with numerator b squared and denominator 4 a squared plus negative c over a

Step 5: Factor the left side into (x + one-half × b)2 and add the right side by getting a common denominator.

The b term is b over a

  • open paren x plus one-half times b over a close paren squared equals the fraction with numerator b squared and denominator 4 a squared plus negative c over a

The common denominator of a and 4a2 is 4a2

  • negative c over a times 4 a over 4 a equals negative 4 a c over 4 a squared

Add the fractions together.

  • open paren x plus b over 2 a close paren squared equals the fraction with numerator b squared and denominator 4 a squared plus negative 4 a c over 4 a squared

Step 6:Take the square root of both sides and get x by itself.

  • the square root of open paren x plus b over 2 a close paren squared equals the square root of the fraction with numerator b squared minus a c and denominator 4 a squared
  • x plus b over 2 a equals plus or minus the fraction with numerator the square root of b squared minus 4 a c and denominator the square root of 4 a squared
  • x plus b over 2 a equals plus or minus the fraction with numerator the square root of b squared minus 4 a c and denominator 2 a

Subtract b over 2 a from both sides. x equals negative b over 2 a plus or minus the fraction with numerator the square root of b squared minus 4 a c and denominator 2 a

Now, combine for one fraction. x equals the fraction with numerator negative b plus or minus the square root of b squared minus 4 a c and denominator 2 a

Quadratic Formula

x equals the fraction with numerator negative b plus or minus the square root of b squared minus 4 a c and denominator 2 a

This is called the Quadratic Formula. The quadratic equation can be used to solve any quadratic equation.

We can now plug straight into this formula to solve quadratic equations.