Quadratic Formula Calculator
How do you use the quadratic formula step by step?
Formula: x = (−b ± √(b² − 4ac)) ÷ 2a Continue your calculation with the Scientific Calculator, or check the Exponent Calculator.
Example: Solve 2x² − 5x − 3 = 0. Here: a=2, b=−5, c=−3.
- Calculate discriminant: D = b² − 4ac = 25 − 4(2)(−3) = 25 + 24 = 49
- Since D = 49 > 0, there are two real roots
- x = (−(−5) ± √49) ÷ (2×2) = (5 ± 7) ÷ 4
- x₁ = (5+7) ÷ 4 = 12 ÷ 4 = 3, x₂ = (5−7) ÷ 4 = −2 ÷ 4 = −0.5
- Verify: 2(3)² − 5(3) − 3 = 18−15−3 = 0 ✓
What does the discriminant tell you?
| Discriminant (D = b²−4ac) | Type of Roots | Example |
|---|---|---|
| D > 0 | Two distinct real roots | x²−5x+6=0: x=2 and x=3 |
| D = 0 | One repeated real root (tangent) | x²−4x+4=0: x=2 (double root) |
| D < 0 | Two complex conjugate roots (no real solution) | x²+1=0: x=±i |
Alternative methods to solve quadratic equations
| Method | When to use | Speed |
|---|---|---|
| Quadratic formula | Always works for any quadratic | Medium |
| Factoring | When roots are integers or simple fractions | Fast (when it works) |
| Completing the square | Deriving the formula; vertex form | Slow |
| Graphing | Approximate roots; visualising | Imprecise |
Real-world applications of quadratic equations
Physics: Projectile motion — height = −16t² + v₀t + h₀. Time to hit ground: solve for t. Finance: Break-even analysis with variable costs. Engineering: Structural beam deflection, circuit analysis. Architecture: Parabolic arch design. Optics: Mirror and lens focal length calculations.
Frequently asked questions
What is the quadratic formula?
x = (−b ± √(b²−4ac)) ÷ 2a. It solves any equation in the form ax²+bx+c=0 where a≠0. The ± gives two solutions (or one if the discriminant is zero). The formula always works — unlike factoring, which only works for equations with integer or simple rational roots.
How do you find the vertex of a parabola?
The vertex x-coordinate: x = −b ÷ (2a). Substitute back to find y. For y = x²−4x+3: vertex x = −(−4) ÷ (2×1) = 2. Vertex y = 4−8+3 = −1. Vertex = (2, −1). The axis of symmetry is x = −b/(2a). The vertex is the minimum (if a>0) or maximum (if a<0) of the parabola.
What are complex roots and when do they occur?
Complex roots occur when the discriminant (b²−4ac) is negative. √(negative number) = imaginary. For x²+4=0: D = 0−16 = −16. x = ±√(−16) ÷ 2 = ±4i ÷ 2 = ±2i. Complex roots always come in conjugate pairs (a+bi and a−bi). They mean the parabola does not cross the x-axis.
Can I factor a quadratic instead of using the formula?
Yes, when the roots are integers: x²−5x+6=0 factors as (x−2)(x−3)=0 giving x=2, x=3. To factor: find two numbers that multiply to ac and add to b. For x²−5x+6: multiply to 6, add to −5 → −2 and −3. This is faster when it works, but the quadratic formula always works regardless of root type.
What is the sum and product of roots?
For ax²+bx+c=0: Sum of roots (x₁+x₂) = −b/a. Product of roots (x₁×x₂) = c/a. These are Vieta's formulas. For 2x²−5x−3=0: Sum = 5/2 = 2.5. Product = −3/2 = −1.5. Check: x₁=3, x₂=−0.5. Sum=2.5 ✓ Product=−1.5 ✓
Sources & References
Sources: Al-Khwarizmi, The Compendious Book on Calculation by Completion and Balancing (c.820 AD); NIST Digital Library of Mathematical Functions; Common Core CCSS.MATH.CONTENT.HSA.REI.B.4.
Vertex form and the parabola
Every quadratic equation y = ax²+bx+c has a parabolic graph. The vertex (turning point) is the most important feature. Vertex x-coordinate: x = −b/(2a). Substitute back to find y. For y = x²−6x+8: vertex x = 6/2 = 3. Vertex y = 9−18+8 = −1. Vertex = (3, −1). Axis of symmetry: x = 3 (the parabola is a mirror image on each side).
| Coefficient | Effect on Parabola | Example |
|---|---|---|
| a > 0 | Opens upward (U shape) — vertex is minimum | y = 2x²+3x+1 (opens up) |
| a < 0 | Opens downward (∩ shape) — vertex is maximum | y = −x²+4x (opens down) |
| |a| large | Narrow parabola | y = 10x² (very steep) |
| |a| small | Wide parabola | y = 0.1x² (very wide) |
| c | y-intercept (where graph crosses y-axis) | y = x²+2x+5 crosses y at 5 |
Physics applications: projectile motion
Projectile motion uses quadratic equations. Height formula: h = −½gt² + v₀t + h₀, where g = 9.81 m/s², v₀ = initial vertical velocity, h₀ = initial height.
Example: A ball thrown upward at 20 m/s from 2m height. When does it hit the ground?
0 = −4.905t² + 20t + 2
Using quadratic formula: a=−4.905, b=20, c=2
D = 400 + 4(4.905)(2) = 400 + 39.24 = 439.24
t = (−20 ± √439.24) / (2 × −4.905) = (−20 ± 20.96) / −9.81
t₁ = (−20 + 20.96) / −9.81 = −0.098s (reject — before throw)
t₂ = (−20 − 20.96) / −9.81 = 4.18 seconds to hit the ground.
Nature of roots: discriminant decision table
| Discriminant (D = b²−4ac) | Roots | Graph Behaviour | Example |
|---|---|---|---|
| D > 0 (positive) | Two distinct real roots | Parabola crosses x-axis at 2 points | x²−5x+6=0: x=2, x=3 |
| D = 0 (zero) | One repeated real root (double root) | Parabola just touches x-axis at 1 point | x²−4x+4=0: x=2 (twice) |
| D < 0 (negative) | Two complex conjugate roots (no real solution) | Parabola never crosses x-axis | x²+x+1=0: x=(−1±i√3)/2 |
Vieta's formulas: sum and product of roots
For ax²+bx+c=0 with roots x₁ and x₂:
Sum of roots: x₁+x₂ = −b/a
Product of roots: x₁×x₂ = c/a
Example: 3x²−7x+2=0. Without solving: sum = 7/3 ≈ 2.33. Product = 2/3 ≈ 0.67.
Check (roots are x=2 and x=1/3): 2+(1/3)=7/3 ✓, 2×(1/3)=2/3 ✓
Vieta's formulas let you quickly verify roots without substituting them back into the equation.
When to factor vs when to use the quadratic formula
Factoring is faster when it works — but only works for equations with integer or simple rational roots. Use the decision guide:
| Situation | Best Method | Why |
|---|---|---|
| Discriminant is a perfect square (D = 0, 1, 4, 9, 16...) | Factor or QF equally fast | Roots will be rational numbers |
| Discriminant is not a perfect square | Quadratic formula | Roots are irrational — factoring won't work |
| Discriminant < 0 | Quadratic formula | Complex roots — only QF handles these |
| Coefficients are large or fractions | Quadratic formula | Avoids trial-and-error |
| a=1 and small integers | Try factoring first | Faster when roots are obvious |
The quadratic formula is the universal solver — it always works. Factoring is the shortcut that works for a subset of equations. In timed exams, use factoring for obvious cases and the formula for everything else.
Completing the square: deriving the quadratic formula
The quadratic formula is derived by completing the square on ax²+bx+c=0. Divide by a: x²+(b/a)x+(c/a)=0. Move constant: x²+(b/a)x = −c/a. Complete the square — add (b/2a)² to both sides: x²+(b/a)x+(b/2a)² = (b/2a)²−c/a. Left side is a perfect square: (x+b/2a)² = (b²−4ac)/4a². Take square root: x+b/2a = ±√(b²−4ac)/2a. Solve for x: x = [−b ± √(b²−4ac)] ÷ 2a. This is the quadratic formula — directly derived from completing the square. Understanding this derivation makes the formula easier to memorise and apply. Related: Square Root Calculator.
The discriminant: nature of roots at a glance
Before solving, the discriminant b²−4ac tells you exactly what kind of roots to expect:
| Discriminant (b²−4ac) | Nature of Roots | Graph Behaviour |
|---|---|---|
| Positive (> 0) | Two distinct real roots | Parabola crosses x-axis at 2 points |
| Perfect square positive | Two rational roots | Can also be solved by factoring |
| Non-square positive | Two irrational roots | Requires quadratic formula or completing the square |
| Zero (= 0) | One repeated real root | Parabola just touches x-axis at vertex |
| Negative (< 0) | No real roots (2 complex roots) | Parabola entirely above or below x-axis |
Quadratic formula in physics: projectile motion
Projectile motion uses quadratic equations to find time of flight, range, and maximum height. The height equation is: h(t) = h₀ + v₀t − ½gt², where h₀ is initial height, v₀ is initial vertical velocity, g = 9.81 m/s².
Worked example: A ball is thrown upward at 20 m/s from a height of 5 m. When does it hit the ground?
- Set h(t) = 0: 0 = 5 + 20t − 4.905t²
- Rearrange: 4.905t² − 20t − 5 = 0 (so a=4.905, b=−20, c=−5)
- Discriminant: (−20)² − 4(4.905)(−5) = 400 + 98.1 = 498.1 > 0 (two real roots)
- t = (20 ± √498.1) ÷ (2 × 4.905) = (20 ± 22.32) ÷ 9.81
- t = 4.32 s (positive root — time to hit ground) or t = −0.237 s (negative, discard)
Factoring vs quadratic formula: when to use which
Factoring works only when roots are integers or simple fractions — and you can spot the factors quickly. The quadratic formula always works. Use this decision guide:
| Situation | Method | Why |
|---|---|---|
| Simple coefficients (a=1, small b,c) | Try factoring first | Faster: find two numbers that multiply to c and add to b |
| Discriminant is a perfect square | Either works | Roots are rational — factoring is possible |
| Discriminant is not a perfect square | Quadratic formula | Irrational roots cannot be factored cleanly |
| Large coefficients (a≠1) | Quadratic formula | Factoring by grouping is tedious; formula is faster |
| Complex (negative discriminant) | Quadratic formula | Formula extends naturally to complex numbers |
Vieta's formulas let you check answers: for roots x₁ and x₂, sum of roots = −b/a and product of roots = c/a. If a=2, b=−7, c=3: roots should sum to 7/2=3.5 and multiply to 3/2=1.5. Solving gives x=3 and x=0.5: sum=3.5 ✓, product=1.5 ✓.