LightFieldStudios/iStock/GettyImages
A quadratic equation contains a single variable raised to the second power. In its standard form, it is expressed as ax² + bx + c = 0, where a, b and c are constants. Unlike linear equations, a quadratic equation always has two solutions, which can be found using one of three methods: factoring, completing the square, or the quadratic formula. The quadratic formula provides a universal solution applicable to any quadratic equation.
For the general quadratic equation ax² + bx + c = 0, the solutions are given by:
\(x = \frac{−b \pm \sqrt{b^2 − 4ac}}{2a}\)
The “±” indicates two distinct solutions: one using the plus sign and the other using the minus sign.
Before applying the formula, ensure the equation is in standard form. If terms appear on both sides of the equation, bring them to one side and combine like terms.
Step 1: Convert to Standard Form
Expand the brackets:
3x² – 12 = 2x² – 2x
Move all terms to the left:
3x² – 2x² + 2x – 12 = 0
Combine like terms:
x² + 2x – 12 = 0
Now the equation is in the form ax² + bx + c = 0 with a = 1, b = 2, c = –12.
Step 2: Plug a, b, and c into the Formula
\(x = \frac{−2 \pm \sqrt{2^2 − 4\times1\times(−12)}}{2\times1}\)
Step 3: Simplify
Compute the discriminant: 4 + 48 = 52
\(x = \frac{−2 \pm \sqrt{52}}{2}\)
Since \(\sqrt{52} \approx 7.21\), we have:
\(x = \frac{−2 + 7.21}{2} \approx 2.61\)
\(x = \frac{−2 − 7.21}{2} \approx −4.61\)
Thus the solutions are x ≈ 2.61 and x ≈ –4.61.
Factoring works best for simple equations where two integers multiply to c and add to b. It becomes challenging when fractional or irrational numbers are involved.
If the equation is in standard form, isolate the quadratic and linear terms, then add (b/2)² to both sides to transform the left side into a perfect square:
\(x^2 + bx + (b/2)^2 = (x + b/2)^2\)
Afterward, solve for x by taking square roots of both sides.
While both methods are valuable, the quadratic formula remains the most reliable technique for all quadratics.
