In this section, we will learn how to find the root(s) of a quadratic equation. Roots are also called x-intercepts or zeros. A quadratic function is graphically represented by a parabola with vertex located at the origin, below the x-axis, or above the x-axis. Therefore, a quadratic function may have one, two, or zero roots.

When we are asked to solve a quadratic equation, we are really being asked to find the roots. We have already seen that completing the square is a useful method to solve quadratic equations. This method can be used to derive the quadratic formula, which is used to solve quadratic equations. In fact, the roots of the function,

f (x) = ax² + bx + c

are given by the quadratic formula. The roots of a function are the x-intercepts. By definition, the y-coordinate of points lying on the x-axis is zero. Therefore, to find the roots of a quadratic function, we set f (x) = 0, and solve the equation,

ax² + bx + c = 0.

We can do this by completing the square as,

Solving for x and simplifying we have,

Thus, the roots of a quadratic function are given by,

This formula is called the quadratic formula, and its derivation is included so that you can see where it comes from. We call the term b² −4ac the discriminant. The discriminant is important because it tells you how many roots a quadratic function has. Specifically, if

We will examine each case individually.

Case 1: No Real Roots

If the discriminant of a quadratic function is less than zero, that function has no real roots, and the parabola it represents does not intersect the x-axis. Since the quadratic formula requires taking the square root of the discriminant, a negative discriminant creates a problem because the square root of a negative number is not defined over the real line. An example of a quadratic function with no real roots is given by,

f(x) = x² − 3x + 4.

Notice that the discriminant of f(x) is negative,

b² −4ac = (−3)²− 4 · 1 · 4 = 9 − 16 = −7.

This function is graphically represented by a parabola that opens upward whose vertex lies above the x-axis. Thus, the graph can never intersect the x-axis and has no roots, as shown below,

Case 2: One Real Root

If the discriminant of a quadratic function is equal to zero, that function has exactly one real root and crosses the x-axis at a single point. To see this, we set b² −4ac = 0 in the quadratic formula to get,

Notice that is the x-coordinate of the vertex of a parabola. Thus, a parabola has exactly one real root when the vertex of the parabola lies right on the x-axis. The simplest example of a quadratic function that has only one real root is,

y = x²,

where the real root is x = 0.

Another example of a quadratic function with one real root is given by,

f(x) = −4x² + 12x − 9.

Notice that the discriminant of f(x) is zero,

b² −4ac = (12)²− 4 · −4 · −9 = 144 − 144 = 0.

This function is graphically represented by a parabola that opens downward and has vertex (3/2, 0), lying on the x-axis. Thus, the graph intersects the x-axis at exactly one point (i.e. has one root) as shown below,

Case 3: Two Real Roots

If the discriminant of a quadratic function is greater than zero, that function has two real roots (x-intercepts). Taking the square root of a positive real number is well defined, and the two roots are given by,

An example of a quadratic function with two real roots is given by,

f(x) = 2x²− 11x + 5.

Notice that the discriminant of f(x) is greater than zero,

b²− 4ac = (−11)²− 4 · 2 · 5 = 121 − 40 = 81.

This function is graphically represented by a parabola that opens upward whose vertex lies below the x-axis. Thus, the graph must intersect the x-axis in two places (i.e. has two roots) as shown below,

*****

In the next section we will use the quadratic formula to solve quadratic equations.

Solving Quadratic Equations