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The Quadratic Formula
The Quadratic Formula
You can solve any quadratic equation by completing the square.
Now we will complete the square to solve ax2 + bx + c = 0. The solutions will be expressed in terms of a, b, and c. These solutions
will give us a formula we can use to solve any quadratic equation.
| Step 1 Isolate the x2-term and the x-term
on one side of the equation.
Subtract c from both sides of the
equation. |
ax2 + bx + c = 0 ax2
+ bx = -c |
| Step 2 If the coefficient of x2 is not 1,
divide both sides of the equation
by the coefficient of x2.
The coefficient of x2 is a.
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| Divide both sides of the equation by a. |
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Step 3 Find the number that completes
the square: Multiply the coefficient
of x by
 . Square the result.
The coefficient of the x-term is
 .
Step 4 Add the result of Step 3 to both
sides of the equation. |
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Add
 to both sides of the
equation.
To combine like terms on the right
side, write both fractions with
denominator 4a2. |
 |
 Combine like terms on the
right side. In the numerator,
write the b2-term first. Step 5 Write the trinomial as the
square of a binomial. Step 6 Finish solving using the
Square Root Property. |
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Use the Square Root Property. Rather
than writing two separate equations, we
write a single equation using the ± sign.
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Subtract
 from both sides and simplify
the radical. |
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Combine the fractions into a single fraction. |
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Note:
If a > 0, then 4a2 = 2a.
If a < 0, then 4a2 = -2a.
So,
The result is called the quadratic formula.
Formula — The Quadratic Formula
The solutions of the quadratic equation ax2 + bx + c = 0 are given
by the quadratic formula:
Here, a, b, and c are real numbers and a ≠
0.
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