Introduction to the Quadratic Formula
The other main approach to solving quadratic equations comes from the fact that x ^{2} = k implies and a technique called completing the square . The solutions to ax ^{2} + bx + c = 0 are
These solutions are abbreviated as
This formula is called the quadratic formula . It will solve every quadratic equation. The quadratic formula is very important in algebra and is worth memorizing. You might wonder why we bother factoring quadratic expressions to solve quadratic equations when the quadratic formula will work. There are two reasons. One, factoring is an important skill in algebra and calculus. Two, factoring is often easier and faster than computing the quadratic formula. The quadratic formula is normally used to solve quadratic equations where the factoring is difficult.
The Quadratic Formula in Practice
Before the formula can be used, the quadratic equation must be in the form ax ^{2} + bx + c = 0. Once a , b , and c are identified, applying the quadratic formula is simply a matter of performing arithmetic.
2 x ^{2} − x − 7 = 0 |
x = 2; b = −1; c = −7 |
10 x ^{2} − 4 = 0 is equivalent to 10 x ^{2} + 0 x − 4 = 0 |
a = 10; b = 0; c = −4 |
3 x ^{2} + x = 0 is equivalent to 3 x ^{2} + x + 0 = 0 |
a = 3; b = 1; c = 0 |
4 x ^{2} = 0 is equivalent to 4 x ^{2} + 0 x + 0 = 0 |
a = 4; b = 0; c = 0 |
x ^{2} + 3 x = 4 is equivalent to x ^{2} + 3 x − 4 = 0 |
a = 1; b = 3; c = −4 |
−8 x ^{2} = −64 is equivalent to 8 x ^{2} + 0 x − 64 = 0 |
a = 8; b = 0; c = − |
Find practice problems and solutions at The Quadratic Formula Practice Problems - Set 1.
Simplifying the Quadratic Formula
The quadratic formula can be messy to compute when any of a , b , or c are fractions or decimals. You can get around this by multiplying both sides of the equation by the least common denominator or some power of ten.
Examples
Example 1:
The fractions in the formula could be eliminated if we multiplied both sides of the equation by 2.
Sometimes the solutions to a quadratic equation need to be simplified.
Example 2:
The denominator is divisible by 2 and each term in the numerator is divisible by 2, so factor 2 from each term in the numerator. Next use this 2 to cancel the 2 in the denominator.
Find practice problems and solutions at The Quadratic Formula Practice Problems - Set 2.
Solving Equations Using the Quadratic Formula
Now that we can identify a , b , and c in the quadratic formula and can simplify the solutions, we are ready to solve quadratic equations using the formula.
Examples
Multiply both sides of the equation by 3 to eliminate the fraction.
0.1 x ^{2} – 0.8 x + 0.21 = 0
Multiply both sides of the equation by 100 to eliminate the decimal.
Find practice problems and solutions at The Quadratic Formula Practice Problems - Set 3.
More practice problems for this concept can be found at: Algebra Quadratic Equations Practice Test.
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