Quadratic Formula

The quadratic formula returns the x-intercept points of a quadratic equation.

Formula

Summary

The quadratic formula calculates the -intercepts of a quadratic equation which forms the shape of a parabola. The general form of a quadratic equation is shown below.

Expression Description
The second-degree coefficient
The first-degree coefficient
The zero-degree coefficient

Usage

The quadratic formula returns the intercepts of a quadratic equation. The formula is useful when a quadratic equation cannot be easily factored. For example, the quadratic equation shown below is difficult to factor and so is a good candidate for the formula.

Applying the quadratic formula to the equation above, we are able to find the roots of the equation. Substituting , and into the formula gives us the expression below.

Then, after some simplification and expanding the (plus-minus symbol) into two epxressions we are left with the intercepts of the equation. Note, as discussed in the explanation below, the formula returns a result that indicates there is no solution.

Examples

Example 1

Quatradic Equation Example

This example demonstrates applying the quadratic formula to the quadratic equation shown in the plot above and in the equation below. The formula returns the places where the parabola crosses the -axis, also called the roots of the equation.

Note, applying the quadratic formula is the same as setting the equation equal to zero and solving for as shown in the expression below.

Steps

  1. Start by setting up the quadratic formula.

  2. Substitute the coefficients of the quadratic formula into equation. In this case, substitute , and into the formula.

  3. Simplify the expression by evaluating arithmetic within the square root operator and in the denominator.

  4. Evaluate the division and square root operator.

  5. Expand the (plus-minus symbol) into two solutions representing the positive and negative result of the square root operator.

    The two roots of the quadratic equation are and .

Example 2

Quatradic Equation Example 2

This example demonstrates applying the quadratic formula to the quadratic equation shown in the plot above and in the equation below. The formula returns the places where the parabola crosses the -axis, also called the roots of the equation.

Note, applying the quadratic formula is the same as setting the equation equal to zero and solving for as shown in the expression below.

Steps

  1. Start by setting up the quadratic formula.

  2. Then substitute the coefficients of the quadratic formula into equation. In this case, substitute , and into the formula.

  3. Simplify the expression by evaluating arithmetic within the square root operator and in the denominator.

  4. Evaluate the division and square root operator.

  5. Expand the (plus-minus symbol) into two solutions representing the positive and negative result of the square root operator.

    The two roots of the quadratic equation are and .

Explanation

The quadratic formula is derived from the general form of the quadratric equation set equal to zero[1]. Visually, a quadratic equation forms the shape of a parabola and the formula returns the points where the parabola intersects with the axis. This is illustrated in the figure below:

The quadratic formula returns the x-intercept points of a quadratic equation.
Figure 1: Quadratic Formula

Plugging in the coefficients: , and of the quadratic equation into the formula will produce two results, because the square root operator returns both a positive and negative solution. This is explicitly shown by expanding the (plus-minus) symbol into two separate expressions.

Edge Cases

Sometimes the quadratic equation does not intersect with the -axis as is the case with the quadratic equation below. When this happens, applying the quadratic formula will result in a negative number as input to the square root operator.

Applying the quadratic formula to this equation gives the following expression. Observe the square-root expression highlighted in red.

A solution can be found if you are willing to expand the domain of input to complex numbers, but otherwise this indicates that there is no solution.

Divide by Zero

The other edge case with this formula is if the input equation isn’t actually quadratic, meaning that the coefficient . In this case, the formula results in a divide by zero error.

Links

References

  1. Derive Quadratic Formula
    Wumbo (internal)