Quadratics don't have to be scary. Learn how to pick the best method (factoring, square roots, completing the square, or the quadratic formula) and solve with confidence.
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Solve Now →A quadratic equation is an equation where the highest power of the variable is 2. The most common way to write it is called standard form:
Here, a, b, and c are numbers and a ≠ 0(if a were 0, it would not be quadratic). Quadratic equations often have two solutions, but they can also have one solution (a repeated root) or no realsolutions.
If you want extra practice and visuals, Khan Academy's Quadratics unit is a solid companion.
If you learn one flowchart for quadratics, learn this one:
Factoring works when you can rewrite the quadratic as a product of two linear factors:
Then you use the zero product property: if the product is 0, at least one factor must be 0. So you set each factor equal to 0 and solve.
Example: x² + 5x + 6 = 0
Factoring is the fastest method, but it doesn't always work cleanly. If you can't factor quickly, don't get stuck—move on to a method that always works.
The square-roots method is perfect when the quadratic has no middle term (or you can isolate x²):
When you take a square root, remember there are two solutions: x = ±√k (plus/minus).
Example: 3x² - 27 = 0
We also have a dedicated guide for this method: how to solve quadratics by square roots.
Completing the square turns a quadratic into a perfect square trinomial, like: (x + m)². This is especially useful when you need vertex form or when you want a method between factoring and the quadratic formula.
Core idea
For x² + bx, add (b/2)² to complete the square.
Example: x² + 6x + 1 = 0
This method takes practice, but it's extremely powerful—especially in more advanced algebra.
If a quadratic does not factor nicely, the quadratic formula will solve it every time. It applies to any equation in standard form ax² + bx + c = 0.
The expression inside the square root, b² − 4ac, is called the discriminant. It tells you how many real solutions you have:
We also wrote a dedicated guide for the formula: how to solve using the quadratic formula.
| If you notice… | Use… | Why… |
|---|---|---|
| It factors quickly | Factoring | Fastest and cleanest |
| No x term (b = 0) | Square roots | Direct and efficient |
| You need vertex form | Completing the square | Builds (x + m)² |
| Nothing factors nicely | Quadratic formula | Always works |
Forgetting the ± when taking square roots
If x² = 9, the solutions are x = 3 and x = -3.
Not writing the equation in standard form first
Most mistakes happen when a, b, and c are chosen incorrectly.
Sign errors with −b or with the discriminant
Use parentheses carefully: (−b ± √(b² − 4ac)) / (2a).
Do quadratics always have two solutions?
Often yes, but sometimes there is one repeated solution (discriminant = 0) or no real solutions (discriminant < 0).
What if I get an ugly decimal?
That's normal for many quadratics. Exact answers may involve radicals or fractions.
How do I know if factoring will work?
If the numbers are small, try it quickly. Otherwise, move on—using the quadratic formula is a smart choice, not a “last resort.”
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