Key takeaway: The quadratic formula solves any equation ax² + bx + c = 0 by giving x = (−b ± √(b² − 4ac)) / (2a), even when the equation does not factor.
Grade 9 – Algebra 1A-REI.B.4bExpressions and Equations
Definition of Quadratic Formula
The quadratic formula is a rule that produces the solutions of every quadratic equation after it has been rearranged to equal zero. You substitute the three coefficients \(a\), \(b\) and \(c\), simplify the part under the root, and finish with two results, one from the plus sign and one from the minus sign.
It works when factoring is difficult or impossible, as with \(x^{2} + 4x – 7 = 0\), and it also tells you in advance how many real solutions to expect: the expression \(b^{2} – 4ac\) under the root is the discriminant. Deriving it once, for letters instead of numbers, is how the formula was built, so each use is just a stored solution being reused.
Formula and notation
Suppose the equation is arranged with \(a\), \(b\) and \(c\) as its coefficients and \(a\) is not 0. Then the quadratic formula states
\[x = \frac{-b \pm \sqrt{b^{2} – 4ac}}{2a}\]
The symbol \(\pm\) gives two answers. The whole numerator is divided by \(2a\), and \(-b\) means the opposite of the coefficient \(b\), including its sign.
Key facts
- Rewrite the equation so that one side is exactly 0 before reading a, b and c.
- A negative number under the root means there are no real solutions.
- Each solution is a place where the matching curve meets the horizontal axis.
- If the root simplifies to a whole number, the equation could have been factored.
Quadratic Formula example
Solve \(5x^{2} + 4x – 12 = 0\) with the quadratic formula.
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Identify \(a = 5\), \(b = 4\), \(c = -12\). Compute \(b^{2} – 4ac = 16 – 4(5)(-12) = 16 + 240 = 256\).
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Substitute: \(x = \frac{-4 \pm \sqrt{256}}{2(5)} = \frac{-4 \pm 16}{10}\).
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Plus sign: \(\frac{12}{10} = \frac{6}{5}\). Minus sign: \(\frac{-20}{10} = -2\).
The solutions are \(x = \frac{6}{5}\) and \(x = -2\).
Common mistake
Watch out
Pablo divides only the square root by \(2a\): \(x = -4 \pm \frac{16}{10}\), and reports \(-2.4\) and \(-5.6\).
Why it is wrong
The fraction bar covers the whole numerator, so \(-4 \pm 16\) must be combined first and then divided by 10. A quick check in the equation exposes the error: \(5(-2)^{2} + 4(-2) – 12 = 0\) works, but \(x = -2.4\) does not.
Related terms and lessons
- Discriminant
- Completing the Square
- Standard Form of a Quadratic
- Zero Product Property
- Zeros of a Function
- Factoring
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