GMAT Quantitative Review

3.0 Math Review

3.2 Algebra

6. Solving Quadratic Equations

The standard form for a quadratic equation is

image,

where a, b, and c are real numbers and image; for example:

image

Some quadratic equations can easily be solved by factoring. For example:

(1)

image

   

(2)

image

A quadratic equation has at most two real roots and may have just one or even no real root. For example, the equation image can be expressed as image, or image; thus the only root is 3. The equation image has no real root; since the square of any real number is greater than or equal to zero, image must be greater than zero.

An expression of the form image can be factored as image.

For example, the quadratic equation image can be solved as follows.

image

If a quadratic expression is not easily factored, then its roots can always be found using the quadratic formula: If image image, then the roots are

image

These are two distinct real numbers unless image. If image, then these two expressions for x are equal to image, and the equation has only one root. If image, then image is not a real number and the equation has no real roots.