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Solve any quadratic equation ax² + bx + c = 0 using the quadratic formula, with the discriminant analysis showing two real roots, a repeated root, or complex conjugate roots.
Math
Generated on May 23, 2026
Solves ax² + bx + c = 0
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Solve any quadratic equation ax² + bx + c = 0 using the quadratic formula, with the discriminant analysis showing two real roots, a repeated root, or complex conjugate roots.
A quadratic equation solver finds the roots of any equation in the form ax² + bx + c = 0 using the quadratic formula. It automatically identifies whether the roots are two distinct real numbers, a single repeated real root, or a complex conjugate pair — based on the sign of the discriminant. Quadratic equations appear all over mathematics, physics, engineering, and finance: projectile motion, parabolic curves, break-even analysis, optimization problems, and many real-world models reduce to quadratic form once the unknowns are set up.
Formula
x = [−b ± √(b² − 4ac)] ÷ (2a). Discriminant D = b² − 4ac.It's the F.Sc Pre-Engineering algebra unit, the test is in three days, and you're staring at an equation like 2x² − 7x − 15 = 0 wondering whether to factor it (slow if you can't see the factors) or apply the quadratic formula (always works, just plug and chug). Honest take: stop trying to factor. The quadratic formula handles every quadratic ever written, and on exam day, the time you save not hunting for factors is the time you have left for the geometry section. Plug a, b, c into x = (−b ± √(b² − 4ac)) / 2a and you're done. The discriminant — that b² − 4ac under the radical — is the part that tells the story before you do any arithmetic. Positive means two real solutions (parabola crosses the x-axis twice, your projectile lands), zero means one repeated root (a perfect touch), negative means complex roots (no real intersection — your projectile never returns to that height). This solver gives you all three cases automatically and shows the discriminant separately so you can sanity-check what kind of answer to expect. Projectile motion, profit-maximization, parabolic-arch engineering, RC circuit analysis — quadratics show up everywhere. The formula has been around since the 9th century when al-Khwarizmi systematized it; nothing has improved on it because nothing needed to.
A quadratic equation solver finds the roots of any equation in the form ax² + bx + c = 0 using the quadratic formula. It automatically identifies whether the roots are two distinct real numbers, a single repeated real root, or a complex conjugate pair — based on the sign of the discriminant. Quadratic equations appear all over mathematics, physics, engineering, and finance: projectile motion, parabolic curves, break-even analysis, optimization problems, and many real-world models reduce to quadratic form once the unknowns are set up.
The quadratic formula is derived by 'completing the square' on ax² + bx + c = 0 — a standard algebraic manipulation that isolates x. The discriminant D = b² − 4ac determines the root type before you even evaluate the square root: D > 0 gives two distinct real roots, D = 0 gives one repeated real root (a double root where the parabola just touches the x-axis), and D < 0 gives two complex conjugate roots (the parabola never crosses the x-axis in the real plane).
Roots of frequently-asked quadratic equations. Shows discriminant type for each case.
| Equation | Discriminant | Root Type | Roots |
|---|---|---|---|
| x² − 5x + 6 = 0 | 1 | Two real | x = 2, x = 3 |
| x² − 7x + 12 = 0 | 1 | Two real | x = 3, x = 4 |
| x² + 2x + 1 = 0 | 0 | One double | x = -1 |
| x² − 4x + 4 = 0 | 0 | One double | x = 2 |
| x² + 2x + 5 = 0 | -16 | Complex | x = -1 ± 2i |
| x² − 2x + 5 = 0 | -16 | Complex | x = 1 ± 2i |
| x² − 9 = 0 | 36 | Two real | x = ±3 |
| 2x² − 3x − 2 = 0 | 25 | Two real | x = 2, x = -0.5 |
Discriminant D = b² − 4ac determines root type: D > 0 → two real, D = 0 → one double, D < 0 → complex conjugate pair.
Vertex of ax² + bx + c is at x = −b/(2a) — useful for finding max/min of parabolas.
Quadratics appear everywhere: projectile motion, profit curves, break-even analysis, parabolic arches.
Complex roots always come in conjugate pairs (a+bi, a−bi) for real-coefficient equations.
x² − 5x + 6 = 0 gives x = 2 and x = 3 (D = 1 > 0, two real roots) — factorable as (x − 2)(x − 3) = 0.
x² − 4x + 4 = 0 gives x = 2 (D = 0, one double root) — the parabola just touches y = 0 and rises.
x² + 2x + 5 = 0 gives x = −1 ± 2i (D = −16 < 0, complex roots) — no real solutions, but complex roots come in conjugate pairs.
Physics projectile motion: the time for a projectile to return to launch height is found by solving a quadratic in t derived from y = v₀t − ½gt².
Business: break-even analysis where revenue and cost are quadratic in production volume often ends up requiring the quadratic formula.
Architecture: parabolic arches and suspension-bridge cable shapes are quadratic — setting cable sag at specific horizontal distances involves solving quadratics.
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