Equations and Inequalities


A quadratic equation is an equation that can be written in the form ax2 + bx + c = 0, where ab, and c are any real numbers with a  0. Any number, x, that satisfies the equation is called a solution or a root of the equation.

Although some quadratic equations can be solved by other, sometimes easier, methods—including factoring—that will be discussed later, every quadratic equation can be solved by using the quadratic formula given in KEY FACT E1.

Key Fact E1


If ab, and c are real numbers with a  0 and if ax bx + c = 0, then

Recall that the symbol ± is read “plus or minus” and that  is an

abbreviation for  or .

As you can see, a quadratic equation has two roots, both of which are determined by the quadratic formula.

The expression b2 – 4ac that appears under the square root symbol is called the discriminant of the quadratic equation. As explained in KEY FACT E2, the dis-criminant provides valuable information about the nature of the roots of a quadratic equation. If we let D represent the discriminant, an alternative way to write the quadratic formula is . The following examples illustrate the proper use of the quadratic formula.

EXAMPLE 12: What are the roots of the equation x2 – 2x – 15 = 0?

EXAMPLE 13: What are the roots of the equation x2 = 10x – 25?

First, rewrite the equation in the form ax2 + bx + c = 0:

Notice that since 10 + 0 = 10 and 10 – 0 = 10, the two roots are each equal to 5. Although some people would say that the equation x2 – 10x + 25 = 0 has only one root, you should say that the equation has two equal roots.

EXAMPLE 14: Solve the equation 2x2 – 4x – 1 = 0.

EXAMPLE 15: Solve the equation x2 – 2x + 2 = 0.

See Chapter 17 for a discussion of the imaginary unit i.

If in the equation ax2 + bx + c = 0, either b or c is equal to 0, it is easier not to use the quadratic formula.

Smart Strategy

Before using the quadratic formula, see if another method would be easier and quicker.

• If b = 0, you can just take a square root.

Note that if c  0, the two solutions will be real. If c > 0, the expression under the square root sign will be negative, and the two solutions will be imaginary.



• If c = 0, you should just factor out an x.

Recall that if a product is equal to 0, one of the factors must be 0. So:



• If you can easily factor the expression ax2 + bx + c, then you can solve the equation ax2 + bx + c = 0 by setting each factor equal to 0.

Factoring is faster than using the quadratic formula if you can immediately determine the factors. However, factoring is a trial and error method. Even if an expression factors, it may take you a long time to do so. Worse, after spending a few minutes trying to factor, you may realize that the expression cannot be factored. Now you have to use the quadratic formula, and it would have taken less time to have done so right away.

Consider the equations 2x2 + 11x – 24 = 0 and 2x2 + 13x – 24 = 0. Only one of them can be solved by factoring. Do you know which one? Probably not. So if you were given one of these equations to solve and you attempted the factoring method, you might waste valuable time in a fruitless effort. Therefore, to solve a quadratic equation, use the quadratic formula unless b = 0 or c = 0 or unless you immediately see how to factor the quadratic expression.

Sometimes a question on a Math 1 test asks for information about the roots of a quadratic equation but does not specifically require you to solve the equation. Looking at the discriminant (D) will allow you to answer such a question.

• If D = 0, as in Example 13, the two roots are  and  . So the two roots are equal. If ab, and c are all rational numbers, then  is also rational.

• If D is negative, as in Example 15, then the equation has no real roots. Both roots are complex (or imaginary) numbers. (See Chapter 17 for a discussion of complex numbers.) If one of the roots is u + vi, the other root is its conjugate: u – vi.

• If D is positive, as in Examples 12 and 14, then the equation has two unequal real roots. If ab, and c are rational, then if D is a perfect square, as in Example 12, the two roots are rational; and if D is not a perfect square, as in Example 14, the two roots are irrational.

The preceding remarks are summarized in the following chart.

Key Fact E2

If ab, and c are rational numbers with a  0, if ax2bx + c = 0, and if D = b2 – 4ac, then

Value of Discriminant

Nature of the Roots

D = 0

2 equal rational roots

D > 0

2 unequal complex roots that are

D > 0


  (i) D is a perfect square

2 unequal rational roots

  (ii) D is not a perfect square   

2 unequal rational roots


Be sure to memorize the facts in this chart.

If ax2 + bx + c = 0 and r1 and r2 are the two roots, then the sum of the roots is

and the product of the roots is

Key Fact E3

If axbx + c = 0, then the sum of the two roots is  and the product of the two roots is .

EXAMPLE 20: Find a quadratic equation for which the sum of the roots is 5 and the product of the roots is 5.

For simplicity, let a = 1. Then , and  . So the equation x– 5x + 5 = 0 satisfies the given conditions.