The Quadratic Formula
Learning Objective(s)
· Write a quadratic equation in standard form and identify the values of a, b, and c in a standard form quadratic equation.
· Use the Quadratic Formula to find all real solutions.
· Use the Quadratic Formula to find all complex solutions.
· Compute the discriminant and state the number and type of solutions.
· Solve application problems requiring the use of the Quadratic Formula.
You can solve any quadratic equation by completing the square—rewriting part of the equation as a perfect square trinomial. If you complete the square on the generic equation ax2 + bx + c = 0 and then solve for x, you find that
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 ax2 + bx + c = 0.
The form ax2 + bx + c = 0 is called standard form of a quadratic equation. Before solving a quadratic equation using the Quadratic Formula, it's vital that you be sure the equation is in this form. If you don't, you might use the wrong values for a, b, or c, and then the formula will give incorrect solutions.
Identify the values of a, b, and c in the standard form of the equation 3x + x2 = 6.
A) a = 3, b = 1, c = 6
B) a = 1, b = 3, c = 6
C) a = 1, b = 3, c = −6
D) a = 3, b = 1, c = −6
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 x2 + bx + c = 0.
· Rewrite the equation so that x2 + bx is isolated on one side.
· Complete the square by adding to both sides.
· Rewrite the perfect square trinomial as a square of a binomial.
· Use the Square Root Property and solve for x.
Can you complete the square on the general quadratic equation ax2 + bx + c = 0? Try it yourself before you continue to the example below. Hint: Notice that in the general equation, the coefficient of x2 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 you 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 using the Quadratic Formula to solve a relatively simple equation first; then you’ll go back and solve it again using another factoring method.
You can check these solutions by substituting 1 and −5 into the original equation.
You get two true statements, so you know that both solutions work: x = 1 or −5. You’ve solved the equation successfully using the Quadratic Formula!
However, upon looking at x2 + 4x = 5, you may have thought “I already know how to do this; I can rewrite this equation as x2 + 4x - 5 = 0, and then factor it as (x + 5)(x - 1) = 0, so x = −5 or 1.” This is correct—and congratulations if you made this connection!
Sometimes, it may be easier to solve an equation using conventional factoring methods, like finding number pairs that sum to one number (in this example, 4) and that produce a specific product (in this example −5) when multiplied. The power of the Quadratic Formula is that it can be used to solve any quadratic equation, even those where finding number combinations will not work.
Most of the quadratic equations you've looked at have two solutions, like the one above. The following example is a little different.
Again, check using the original equation.
Let's try one final example. This one also has a difference in the solution.
Check these solutions in the original equation. Be careful when expanding the squares and replacing i2 with -1.
Use the Quadratic Formula to solve the equation x2 - 2x - 4 = 0.
A) x = 2
B) x =11, x = −9
C) ,
D) ,
These examples have shown that a quadratic equation may have two real solutions, one real solution, or two complex solutions.
In the Quadratic Formula, the expression underneath the radical symbol determines the number and type of solutions the formula will reveal. This expression, b2 - 4ac, is called the discriminant of the equation ax2 + bx + c = 0.
Let’s think about how the discriminant affects the evaluation of , and how it helps to determine the solution set.
· If b2 - 4ac > 0, then the number underneath the radical will be a positive value. You can always find the square root of a positive, so evaluating the Quadratic Formula will result in two real solutions (one by adding the positive square root, and one by subtracting it).
· If b2 - 4ac = 0, then you will be taking the square root of 0, which is 0. Since adding and subtracting 0 both give the same result, the "±" portion of the formula doesn't matter. There will be one real solution.
· If b2 - 4ac < 0, then the number underneath the radical will be a negative value. Since you cannot find the square root of a negative number using real numbers, there are no real solutions. However, you can use imaginary numbers. You will then have two complex solutions, one by adding the imaginary square root and one by subtracting it.
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.
B) The equation has one solution.
C) The equation has zero solutions.
Quadratic equations are widely used in science, business, and engineering. Quadratic equations are commonly used in situations where two things are multiplied together and they both depend on the same variable. For example, when working with area, if both dimensions are written in terms of the same variable, you use a quadratic equation. Because the quantity of a product sold often depends on the price, you sometimes use a quadratic equation to represent revenue as a product of the price and the quantity sold. Quadratic equations are also used when gravity is involved, such as the path of a ball or the shape of cables in a suspension bridge.
A very common and easy-to-understand application is the height of a ball thrown at the ground off a building. Because gravity will make the ball speed up as it falls, a quadratic equation can be used to estimate its height any time before it hits the ground. Note: The equation isn't completely accurate, because friction from the air will slow the ball down a little. For our purposes, this is close enough.
The area problem below does not look like it includes a Quadratic Formula of any type, and the problem seems to be something you have solved many times before by simply multiplying. But in order to solve it, you will need to use a quadratic equation.
A ball is launched upward at 48 feet/second from a platform that is 100 feet high. The equation giving its height t seconds after launch is h = −16t2 + 48t + 100. The ball will shoot up to 136 feet high, then begin to come back down. About how long will the ball take to get to that maximum height?
A) 1.5 seconds
B) 3.6 seconds
C) 4.4 seconds
D) This problem cannot be solved.
Quadratic equations can appear in different applications. The Quadratic Formula is a useful way to solve these equations, or any other quadratic equation! The Quadratic Formula, .
The discriminant of the Quadratic Formula is the quantity under the radical, . It determines the number and the type of solutions that a quadratic equation has. If the discriminant is positive, there are 2 real solutions. If it is 0, there is 1 real solution. If the discriminant is negative, there are 2 complex solutions (but no real solutions).
