What if you had a quadratic equation like that you could not easily factor? How could you use its coefficient values to solve it? After completing this Concept, you'll be able to use the quadratic formula to solve equations like this one.
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CK-12 Foundation: 1008S The Quadratic Formula
For more examples of solving quadratic equations using the quadratic formula, see the Khan Academy video at
Guidance
The Quadratic Formula is probably the most used method for solving quadratic equations. For a quadratic equation in standard form, , the quadratic formula looks like this:
This formula is derived by solving a general quadratic equation using the method of completing the square that you learned in the previous section.
We start with a general quadratic equation:
Subtract the constant term from both sides:
Divide by the coefficient of the term:
Rewrite:
Add the constant to both sides:
Factor the perfect square trinomial:
Simplify:
Take the square root of both sides:
Simplify:
This can be written more compactly as .
You can see that the familiar formula comes directly from applying the method of completing the square. Applying the method of completing the square to solve quadratic equations can be tedious, so the quadratic formula is a more straightforward way of finding the solutions.
Solve Quadratic Equations Using the Quadratic Formula
To use the quadratic formula, just plug in the values of and .
Example A
Solve the following quadratic equations using the quadratic formula.
a)
b)
c)
Solution
Start with the quadratic formula and plug in the values of and .
a)
Answer: and
b)
Answer: and
c)
Answer: and
Often when we plug the values of the coefficients into the quadratic formula, we end up with a negative number inside the square root. Since the square root of a negative number does not give real answers, we say that the equation has no real solutions. In more advanced math classes, you’ll learn how to work with “complex” (or “imaginary”) solutions to quadratic equations.
Example B
Use the quadratic formula to solve the equation .
Solution
Answer: There are no real solutions.
To apply the quadratic formula, we must make sure that the equation is written in standard form. For some problems, that means we have to start by rewriting the equation.
Finding the Vertex of a Parabola with the Quadratic Formula
Sometimes a formula gives you even more information than you were looking for. For example, the quadratic formula also gives us an easy way to locate the vertex of a parabola.
Remember that the quadratic formula tells us the roots or solutions of the equation . Those roots are , and we can rewrite that as
Recall that the roots are symmetric about the vertex. In the form above, we can see that the roots of a quadratic equation are symmetric around the coordinate , because they are units to the left and right (recall the sign) from the vertical line .
Example C
In the equation , the roots -1 and 3 are both 2 units from the vertical line , as you can see in the graph below:
Watch this video for help with the Examples above.
CK-12 Foundation: 1008 The Quadratic Formula
Vocabulary
- For a quadratic equation in standard form, , the quadratic formula looks like this:
- The quadratic formula tells us the roots or solutions of the equation . Those roots are , and we can rewrite that as
- The roots are symmetric about the vertex . In the form above, we can see that the roots of a quadratic equation are symmetric around the coordinate , because they are units to the left and right (recall the sign) from the vertical line .
Guided Practice
Solve the following equations using the quadratic formula.
a)
b)
Solution
a)
Answer: and
b)
Answer: no real solutions
Practice
Solve the following quadratic equations using the quadratic formula.