Solving Quadratic Equations Using the Quadratic Formula

 

Learning Objective(s)

·         Solve quadratic equations using the quadratic formula.

 

 

We can solve any quadratic equation by completing the square—turning a polynomial into a perfect square trinomial. If we complete the square on the generic equation  and then solve for x, we find that .  This rather awkward looking equation is known as the quadratic formula.

 

This formula is very helpful for solving quadratic equations that are difficult or impossible to factor, and using it can be faster than completing the square. The quadratic formula can be used to solve any quadratic equation of the form .

 

 

Let's complete the square on the general equation, , and see exactly how that produces the quadratic formula. Recall the process of completing the square:

 

·         Start with an equation of the form .

 

·         Rewrite the equation so that  is isolated on one side.

·         Complete the square by adding to both sides.

·         Rewrite as a square of a binomial and solve for x.

 

Can you complete the square on the general quadratic equation ? Try it yourself before you continue to the example below.  Hint: When working with the general equation , there is an added complication, in that the coefficient of  is not equal to 1. You can divide the equation by a, which makes some of the expressions a bit messy, but if you are careful, everything will work out, and at the end, you’ll have the quadratic formula!

 

 

There we have it, the quadratic formula.

 

 

The quadratic formula will work with any quadratic equation, but only if the equation is in standard form, . To use it, follow these steps:

 

·         Put the equation in standard form first.

 

·         Identify the coefficients, a, b, and c. Be careful to include negative signs if the bx or c terms are subtracted.

 

·         Substitute the values for the coefficients into the quadratic formula.

 

·         Simplify as much as possible.

 

·         Use the ± in front of the radical to separate the solution into two values: one in which the square root is added, and one in which it is subtracted.

 

·         Simplify both values to get the possible solutions.

 

That's a lot of steps. Let’s try them:

 

 

 

The solution to the quadratic equation gives the x-coordinates of the x-intercepts, or the roots of a quadratic equation. The roots of the quadratic equation are the values where the parabola crosses the x-axis. We can check this by looking at the graph of the function  and see that the roots are (4, 0) and (, 0).

 

 

 

The example above shows a quadratic equation with two solutions. Below is an example with one solution. Compare the simplified radicals of the two examples:

 

 

 

This quadratic equation has only one solution, so the graph of the function  will touch the x-axis once. It has only one root.

 

 

 

Something to notice—the quadratic equation  can be factored to .  So although the quadratic formula gave us the solution, it would have been faster to factor it. It's worth checking to see if a quadratic equation can be easily factored before applying the quadratic formula.

 

 

Use the quadratic formula to solve the equation .   A) x = 2 B) x =11, x = -9 C) , D) ,   Show/Hide Answer A) Incorrect. This answer could be reached by incorrectly factoring to . a = 1, b = -2, and c = -4. So. The answer is  or .     B) Incorrect. . If you ignore the square root, this becomes x = 11 or x = –9, so you may have forgotten to take the square root of 20. , so the correct answer is  or .   C) Incorrect. Using the formula, . If you forget that the denominator is under both terms in the numerator, you might get  or . However, the correct answer is , so  or .   D) Correct. Using the formula, , so  or .

 

A quadratic equation may have two roots, one root, or no roots. In the quadratic formula, the expression underneath the radical symbol determines how many solutions the formula will have. This expression, , is called the discriminant of the equation .

 

Let’s think about how  will affect the evaluation of , and how it helps to determine the solution set.

 

·         If , then the number underneath the radical will be a positive value. We can always find the square root of a positive number, so evaluating the quadratic formula will result in two solutions (one by adding the positive square root, and one by subtracting it).

 

·         If , then we will be taking the square root of 0, and the “±” term drops out of the evaluation of the quadratic formula. (Adding zero and subtracting zero will both give the same result.) There will be one solution.

 

·         If , then the number underneath the radical will be a negative value. Since we cannot find the square root of a negative number (at least not using the real number system), we cannot evaluate the formula any further. There will be no solutions.

 

 

 

Suppose a quadratic equation has a discriminant that evaluates to zero. Which of the following statements is always true?   A) The equation has two solutions and the graph of the parabola will open upward. B) The equation has one solution and the graph of the parabola will open downward. C) The equation has one solution and you can’t tell anything about the direction of the parabola. D) The equation has zero solutions and you can’t tell anything about the direction of the parabola.   Show/Hide Answer A) Incorrect. A discriminant of zero means the equation has one solution, and it tells you nothing about direction that the graph of the parabola opens. (The value of the coefficient a is what tells you the direction.)   B) Incorrect. A discriminant of zero does show the equation has one solution, but it provides no information about the direction of the parabola (you need the value of the coefficient a for that).   C) Correct. A discriminant of zero means the equation has one solution. You need the value of the coefficient a to tell you what direction the parabola opens.   D). Incorrect. When the discriminant is zero, the parabola will have one solution. However, you are correct that you can’t tell anything about the direction of the parabola. You need to know the value of the coefficient a to determine the direction.

 

The quadratic formula, , is found by completing the square of the quadratic equation .  The formula can be used to find the solution to a quadratic equation and to identify any possible roots, or x-intercepts, of the function.

 

The discriminant of the quadratic formula is the quantity under the radical, . It determines how many solutions there are to the quadratic equation. If the discriminant is positive, there are 2 roots. If it is zero, there is 1 root. If the discriminant is negative, there are 0 roots.