QuickCalculators solves ax squared plus bx plus c equals zero with the quadratic formula, labels the discriminant case, and writes exact roots when radicals simplify. Enter coefficients a, b, and c, then read the two roots, the discriminant value, and the complex or repeated form when that case applies.
Solve a quadratic equation with the quadratic formula
A quadratic equation is a degree-two polynomial set equal to zero. The quadratic formula returns the roots for any a not equal to zero by combining the coefficients into one expression under a square root. QuickCalculators substitutes the typed a, b, and c, then reports the roots in decimal and exact form when the radical simplifies cleanly.
The formula applies whether the parabola crosses the x-axis twice, once, or not at all in the reals. Complex conjugate roots appear when the discriminant is negative. A repeated root appears when the discriminant is zero. The work panel shows each substituted value so the arithmetic stays checkable.
Apply the quadratic formula
The quadratic formula is x equals negative b plus or minus the square root of b squared minus 4ac, all over 2a. The minus b term sits in the numerator with the plus-or-minus square root, and the whole fraction divides by twice a. QuickCalculators evaluates the discriminant first, then splits into the plus branch and the minus branch.
Order of operations matters: square b, multiply 4ac, subtract, take the square root of that result, then combine with negative b before dividing by 2a. Skipping the absolute grouping around the numerator produces wrong signs. The calculator keeps those steps in sequence in the show-work list.
Read the discriminant
The discriminant is b squared minus 4ac. A positive discriminant yields two distinct real roots. A zero discriminant yields one repeated real root. A negative discriminant yields a conjugate pair of complex roots written with the imaginary unit. QuickCalculators prints the discriminant value and names that case beside the roots.
For a equals 1, b equals negative 5, and c equals 6, the discriminant is 25 minus 24, or 1, so two real roots exist. For a equals 1, b equals 2, and c equals 5, the discriminant is 4 minus 20, or negative 16, so the roots are complex. That case label prevents treating a negative square root as a blank result.
Handle real and complex roots
Real roots include simplified radicals such as 1 plus or minus the square root of 3 from x squared minus 2x minus 2 equals 0. Complex roots use the imaginary unit, as in negative 1 plus or minus 2i from x squared plus 2x plus 5 equals 0. QuickCalculators writes conjugate pairs so both members stay visible.
Repeated roots list the same real value twice, matching the double root of a perfect-square trinomial. Exact radical forms stay preferred when the discriminant factors into a perfect square times a square-free part. Decimal approximations sit beside those exact strings for graphing or calculator checks.
Solve x squared minus 5x plus 6 equals 0
The worked case uses a equals 1, b equals negative 5, and c equals 6. The discriminant is 1. The roots are 2 and 3, which factor as (x minus 2)(x minus 3) equals zero and match the engine fixture on QuickCalculators.
- Compute the discriminant: 25 minus 24 equals 1.
- Apply plus or minus: (5 plus 1) / 2 equals 3, and (5 minus 1) / 2 equals 2.
- Report roots x equals 2 and x equals 3.
That pair is the reference solution for this page.
Compare the formula to completing the square
Completing the square rewrites the same quadratic by adding and subtracting a constant so a perfect square appears, then taking square roots. The quadratic formula packages those algebra steps into one plug-in expression. Both methods return the same roots for the same coefficients.
QuickCalculators focuses on the formula path on this page. Completing the square remains a sibling method for the same problem type and appears in teaching notes without requiring a separate live link here. Use whichever method the assignment names; the numeric roots should agree.
Avoid this common mistake
The misconception named "divide only by 2, not by 2a" drops the coefficient a from the denominator. When a is not 1, that shortcut shifts both roots. Always divide by twice a after combining the numerator. The work panel shows 2a explicitly so that slip is harder to miss.
Frequently asked questions
What is the quadratic formula?
The quadratic formula is x equals negative b plus or minus the square root of b squared minus 4ac, all over 2a. It solves ax squared plus bx plus c equals zero for a not equal to zero. QuickCalculators evaluates that expression for the entered coefficients.
How do you solve a quadratic equation with the quadratic formula?
To solve with the quadratic formula, identify a, b, and c, compute the discriminant, then evaluate both plus and minus branches over 2a. The calculator performs those substitutions and lists the steps. Enter the three coefficients in the form fields to run the same path.
What does the discriminant tell you?
The discriminant tells whether the roots are two real numbers, one repeated real number, or a complex conjugate pair. Positive, zero, and negative values map to those three cases in order. The result panel names the case next to the discriminant value.
What are the roots of x squared minus 5x plus 6 equals 0?
The roots of x squared minus 5x plus 6 equals 0 are 2 and 3. The discriminant is 1, and both branches over 2a land on those integers. Entering a equals 1, b equals negative 5, and c equals 6 returns the same pair.
Can the quadratic formula give complex roots?
The quadratic formula can give complex roots when the discriminant is negative. The square root of a negative number is written with the imaginary unit, and the roots form a conjugate pair such as negative 1 plus or minus 2i. QuickCalculators prints that a plus or minus bi form for the complex case.
When is there only one solution?
There is only one distinct real solution when the discriminant is zero. That repeated root still satisfies the equation twice in the algebraic sense. The calculator labels the repeated-root case and shows the single real value.
Why must a be nonzero?
The coefficient a must be nonzero because a zero leading term would make the equation linear, not quadratic, and the formula divides by 2a. Entering a equals 0 returns an error instead of a false quadratic solve. Use a linear solve path when the degree drops.
Summary
QuickCalculators solves ax squared plus bx plus c equals zero with the quadratic formula and reports discriminant cases for real, repeated, and complex roots. Exact forms include 1 plus or minus the square root of 3, and negative 1 plus or minus 2 times the imaginary unit.
The fixture x squared minus 5x plus 6 equals 0 yields roots 2 and 3. Always divide by 2a, and treat a zero leading coefficient as invalid for this formula.