HesapHanem

Discriminant Calculator

Enter the coefficients of your quadratic equation to find its discriminant and root count.

Result

49.00
2

How it is calculated

For a quadratic ax² + bx + c = 0 the tool computes Δ = b² − 4ac and shows it with two decimals. It then reports the number of real roots from the sign of that displayed value: Δ > 0 gives 2, Δ = 0 gives 1 (a repeated root), Δ < 0 gives 0. The coefficient a cannot be zero, because the equation would no longer be quadratic, so that input returns no result; b and c may be zero or negative. Because the count follows the rounded value, a true Δ smaller than 0.005 in size (for example 0.0004 with a = 1, b = 0.1, c = 0.0024) displays as 0.00 and is reported as 1 root, so very small decimal coefficients deserve a manual check. The roots themselves are not calculated.

Worked example

Take 2x² + 3x − 5 = 0, so a = 2, b = 3, c = −5. Step 1: b² = 9. Step 2: 4ac = 4 × 2 × (−5) = −40. Step 3: Δ = 9 − (−40) = 49.00. Because Δ is positive the tool reports 2 real roots. Since 49 is a perfect square, those roots are also rational numbers.

Try this example →

Common mistakes

  • Dropping the sign of c. With c = −5 the term 4ac is negative, so subtracting it adds 40; forgetting this turns 49 into −31.
  • Entering the equation before moving every term to one side. The equation must be in the form ax² + bx + c = 0 first.
  • Reading "1 real root" as "no solution". Δ = 0 means one repeated root, the vertex touching the x-axis.
  • Reading "0 real roots" as "no solution at all". It means no real solution; the equation still has two complex roots.
  • Entering 0 for a, which describes a straight line, not a parabola; the tool shows no result.

About this calculator

The discriminant of a quadratic equation ax² + bx + c = 0 is Δ = b² − 4ac, the expression that sits under the square root in the quadratic formula. Its sign alone tells you how many real solutions the equation has without solving it fully: a positive discriminant means two distinct real roots, a discriminant of exactly zero means one repeated real root (the parabola just touches the x-axis), and a negative discriminant means no real roots, only a pair of complex conjugate solutions.

Algebra and precalculus students and teachers use the discriminant to quickly check how many times a parabola crosses the x-axis before graphing it, and it shows up again in physics and engineering whenever a quadratic model — projectile motion, circuit equations, optimization problems — needs to be checked for whether a real solution exists at all.

From a, b and c it computes b² − 4ac and tells you whether the quadratic has two real roots, one repeated root or none.

Was this helpful?

You can pick up to 3 reactions.

Comments

  • Be the first to comment.

Popular calculators

Loan Calculator

Monthly payment, total interest and full amortization schedule.

Percentage Calculator

Percent change, percent of, and ratio calculations.

Discount Calculator

Calculate final price, savings, and discount amount.

BMI Calculator

Body mass index from height and weight (metric or imperial).

Age Calculator

Compute your exact age in years, months and days from a birth date.

Compound Interest Calculator

Calculate the final amount and interest earned with compounding.

All Calculators