Quadratic Equation - Standard Form

 

You can solve any quadratic equation by completing the square rewriting part of the equation as a perfect square trinomial. If you complete the square on the generic equation ax2 + bx + c = 0 and then solve for x, you find that This equation is known as the Quadratic Formula.

 

This formula is very helpful for solving quadratic equations that are difficult or impossible to factor, and using it can be faster than completing the square. The Quadratic Formula can be used to solve any quadratic equation of the form ax2 + bx + c = 0.

 

The form ax2 + bx + c = 0 is called standard form of a quadratic equation. Before solving a quadratic equation using the Quadratic Formula, it's vital that you be sure the equation is in this form. If you don't, you might use the wrong values for a, b, or c, and then the formula will give incorrect solutions.

 

Example:

Rewrite the equation 3x + 2x2 + 4 = 5 in standard form and identify a, b, and c.

3x + 2x2 + 4 = 5

3x + 2x2 + 4 – 5 = 5 – 5

 

First be sure that the right side of the equation is 0. In this case, all you need to do is subtract 5 from both sides.

3x + 2x2 – 1 = 0

2x2 + 3x – 1 = 0

 

Simplify, and write the terms with the exponent on the variable in descending order.

2x2

+

3x

1

=

0

 

 

 

 

ax2

 

bx

 

c

   

a = 2, b = 3, c = −1

 

Now that the equation is in standard form, you can read the values of a, b, and c from the coefficients and constant. Note that since the constant 1 is subtracted, c must be negative.

Answer:

2x2 + 3x – 1 = 0; a = 2, b = 3, c = −1