Solve ax² + bx + c = 0 using the discriminant and quadratic formula, including repeated, real and complex roots.
Choose an example or type the known values, then calculate to see the answer and working.
A quadratic equation contains a non-zero x-squared term and can have two real roots, one repeated real root or a pair of complex roots. This calculator evaluates the discriminant before applying the quadratic formula. That sequence explains the type of answer instead of displaying roots without context. Enter the signed coefficients from standard form ax² + bx + c = 0.
Use this page after moving every term to one side and collecting like powers of x. It is useful when factorisation is not obvious or when you want to verify factored roots. The leading coefficient a cannot be zero. If the course has not introduced complex numbers, a negative discriminant should be interpreted as no real roots rather than copied without explanation.
The equation must equal zero before a, b and c are identified. Moving a term across equality changes its sign. A missing x term means b = 0; it must not shift the position of c.
A positive discriminant gives two distinct real roots, zero gives one repeated root and a negative value gives complex roots. Checking D first catches many sign and coefficient errors.
The plus-minus symbol represents two calculations when the square-root term is non-zero. Reporting only the plus branch loses a root and may miss a second intercept or valid context value.
| Mistake | Why it fails | Better check |
|---|---|---|
| Entering b without its sign | For x² - 5x + 6, b is -5, not 5. | Read coefficients from standard form and include every sign. |
| Forgetting the entire denominator | The expression divides the complete numerator by 2a. | Use brackets around -b ± √D before dividing. |
| Rejecting a repeated root | When D = 0, both branches produce the same value, which is still a valid root. | Report it as one repeated root and verify it by substitution. |
Use a = 1, b = -5 and c = 6 from the standard-form equation.
Estimate the sign, scale, range or likely form of the answer before calculating. A prediction makes an input error easier to notice.
Calculate D = (-5)² - 4(1)(6) = 25 - 24 = 1.
Substitute into the formula: x = (5 ± √1) ÷ 2.
The two branches give x = (5 + 1) ÷ 2 = 3 and x = (5 - 1) ÷ 2 = 2.
Check by factorisation or substitution: (x - 2)(x - 3) expands to x² - 5x + 6.
Factorisation is often faster when integer factors are visible. The quadratic formula is general and provides a useful independent check.
The graph has no real x-intercepts. The calculator reports complex roots using i, subject to the learner's course scope.
Keep exact surd or fractional form when required. If a decimal is needed, round only the final roots to the stated precision.
The result points to related practice, but the best next step depends on what the learner could explain without help.