Quadratic formula explained with examples
Use this page when the quadratic does not factor neatly or when you want one method that works even when brackets are not obvious.
Start here if you want the short version before reading the full method.
- The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a.
- It is most useful when a quadratic does not factor easily or when you want a general method that always applies.
- Before substituting, identify a, b, and c from ax^2 + bx + c = 0 carefully, including signs.
What this topic means and what to look for first.
This method is often the safest fallback because it does not depend on spotting factor pairs.
Most errors happen before the arithmetic even begins: the real risk is misreading a, b, or c, or losing the ± sign in the middle of the formula.
One reliable route through the topic.
- 1Rewrite the quadratic in the form ax^2 + bx + c = 0 if it is not already there.
- 2Identify a, b, and c, keeping the signs exactly as they appear.
- 3Substitute into the quadratic formula and simplify the discriminant first.
- 4Take the square root carefully, remembering the ± branch gives two possible roots.
- 5Simplify the final answers and check them in the original equation.
See the method in action.
2x^2 + 3x - 2 = 0
- Here a = 2, b = 3, and c = -2.
- Substitute to get x = (-3 ± √(9 + 16)) / 4.
- This simplifies to x = (-3 ± 5) / 4, so the roots are x = 1/2 and x = -2.
x^2 + 4x + 1 = 0
- Here a = 1, b = 4, and c = 1.
- Substitute to get x = (-4 ± √(16 - 4)) / 2.
- This simplifies to x = (-4 ± √12) / 2 = -2 ± √3.
x^2 - 6x + 9 = 0
- Here a = 1, b = -6, and c = 9.
- The discriminant is (-6)^2 - 4(1)(9) = 36 - 36 = 0.
- A discriminant of 0 means one repeated root, so x = 3.
Things that commonly send the method off track.
- Reading the sign of b incorrectly when the middle term is negative.
- Calculating b^2 - 4ac incorrectly, especially when c is negative.
- Forgetting the ± and therefore losing one of the roots.
- Simplifying the square root or denominator inconsistently in the final step.
Use a short verification pass before moving on.
- Substitute each root back into the original equation and confirm the left side becomes 0.
- Compare the discriminant with the result: positive should give two real roots, zero gives one repeated root, and negative means no real roots.
Try a few variations before switching to a calculator or solver tool.
- x^2 + 2x - 7 = 0
- 3x^2 + x - 4 = 0
- x^2 - 10x + 25 = 0
- 2x^2 + 4x + 5 = 0
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Short answers worth checking.
Use it when the quadratic does not factor cleanly or when you want one general method that works for every quadratic equation.
The discriminant b^2 - 4ac tells you how many real roots the quadratic has and whether they repeat.
The ± part of the formula creates two possible values whenever the square root term is not zero.
Continue with the next closely related topic.
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CureMath uses artificial intelligence to suggest how a maths problem could potentially be solved. AI can make mistakes.
Check important answers independently before relying on them.