9.4 - Quadratic equations

Before reading this section you may want to review the following topics:

In this section we discuss four methods of solving quadratic equations:

graphing,
factoring,
completing the square, and
using the quadratic formula.

1. Solving quadratic equations by graphing

To solve the quadratic equation a x² + b x + c = 0, replace the zero on the right-hand-side with the variable y and graph the resulting quadratic function y = a x² + b x + c. The graph is a parabola. The points where the parabola crosses the x axis are the points where y = 0 and hence are the roots or solutions of the quadratic equation.

This method is also the basis of computer methods used to solve more complicated equations.

Example: Solve the quadratic equation x² − 3 x + 2 = 0.

Solution: Replace 0 with y to create the corresponding quadratic function

y = x² − 3 x + 2

and draw its graph. The parabola crosses the x axis at x = 1 and x = 2. This means that the solutions of the quadratic equation x² − 3 x + 2 = 0 are x = 1 and x = 2.

2. Solving quadratic equations by factoring

The material in this section is based on the following topics, which you may want to first review:

factoring quadratic expressions
solving equations of the form a · b = 0

If the expression on the left-hand-side of the quadratic equation a x² + b x + c = 0 can be factored like this:

(x − x₁)(x − x₂) = 0,

then the solutions are x = x₁ and x = x₂. When you can spot the factors, this is probably the easiest of the four methods.

Example: Solve the quadratic equation x² + 3 = 4 x by factoring.

Solution: You must first put the quadratic equation into the standard form

x² − 4 x + 3 = 0.

The left-hand-side can be factored:

(x − 1)(x − 3) = 0.

Therefore the solutions of the quadratic equation are x = 1 and x = 3.

Warning: A common error is to think that the solutions are −1 and −3. This is not correct; the solutions are the values of x that make the factors vanish (become equal to zero).

3. Solving quadratic equations by completing the square

The quadratic equation a x² + b x + c = 0 can be solved for x by completing the square. Here are the steps:

Transpose (add or subtract) the quadratic and linear terms to the left-hand-side of the equation and the constant term to the right-hand-side:
a x² + b x = −c
Divide both sides of the equation by a so that the coefficient of the quadratic term is equal to 1:
Take the coefficient of the linear term (namely b/a), divide it by 2, and then square it. This produces the quantity . Add this quantity to both sides of the equation. This turns the left hand side into a perfect square:
Now factor the left hand side
and combine the fractions on the right hand side:
Notice that this is now an equation in which the unknown occurs just once. This is the key result of the completing the square method. Such an equation can always be solved by simply inverting the operations that were applied to the unknown, one at a time, starting with the last one, until x is isolated.

The first thing to do is to take the square root of both sides. This gives 2 equations, one with a + sign on the right hand side, the other with a − sign (this is indicated by the ± symbol):
Next subtract from both sides:
This equation has a nice geometric interpretation. The two values of x are indicated by the red dots. The first term of the equation locates the midpoint (the point halfway between them) and the second term (the one after the ± symbol) gives the distance from the midpoint to either one of them:
It is customary to combine the fractions on the right hand side of the equation. The result is known as the quadratic formula:

Example: Solve the quadratic equation x² − 4 x + 3 = 0 by completing the square.

Solution: Here are the steps:

Move the constant term to the right-hand-side:
x² − 4 x = −3.
The coefficient of the linear term is −4. Taking half of it and squaring that gives 4. Add this to both sides of the equation:
x² − 4 x + 4 = −3 + 4.
This makes the left-hand-side a perfect square; rewrite it in factored form:
(x − 2)² = 1.
Take the square root of both sides. This leads to two equations as indicated by the ± sign on the right-hand-side.
x − 2 = ±1.
Adding 2 to both sides gives the two solutions x = 1 and x = 3.

4. Solving quadratic equations by quadratic formula

The solutions of the quadratic equation a x² + b x + c = 0 are given by the quadratic formula:

Note:

The quadratic formula was derived by completing the square.
There are two solutions. The + and − signs each give one of them.
The quantity b² − 4 a c under the square root is called the discriminant.
If the discriminant is negative, then the solutions are not real numbers (they are complex numbers) and the parabola y = a x² + b x + c does not cross the x axis.
If the discriminant is zero, then the two solutions are equal (we also say that there is one “double” root) and the parabola just touches the x axis.

Example: The solutions of the quadratic equation x² − 4 x + 3 = 0 can be found using the quadratic formula:

with a = 1, b = −4 and c = 3:

The + sign gives the solution x = 3 and the − sign gives the solution x = 1.

Algebra Coach Exercises

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