Quadratic Equation Solver

Find the real and complex roots of any quadratic equation (ax² + bx + c = 0). Get step-by-step discriminant evaluation (Δ), quadratic formula breakdown, vertex coordinates, and parabolic plot.

What is a Quadratic Equation? (Intuition & Parabolic Roots)

A quadratic equation is a second-order polynomial equation of the standard form ax2 + bx + c = 0, where a ≠ 0. When plotted on a Cartesian coordinate plane, a quadratic equation forms a smooth, symmetrical U-shaped curve known as a parabola. The solutions (also called roots, zeros, or x-intercepts) represent the exact points where the parabola crosses or touches the horizontal x-axis.

If a soccer ball is kicked into the air, its trajectory follows a downward-opening parabola governed by gravity. The roots of the quadratic equation indicate the exact moment the ball launches (t = 0) and the exact second it lands back on the turf. CalcSolver's Quadratic Equation Solver evaluates real and complex roots, displays the discriminant Δ, locates the vertex coordinate (h, k), and details each step using factoring, completing the square, and the quadratic formula.

The Quadratic Formula & The Discriminant (Δ)

The universal algebraic formula for solving any quadratic equation is:

Quadratic Formula: x = [-b ± √(b2 - 4ac)] / (2a)
The Discriminant (Δ): Δ = b2 - 4ac
• If Δ > 0: Two distinct real roots (parabola crosses x-axis twice)
• If Δ = 0: Exactly one repeated real root (parabola touches x-axis at vertex)
• If Δ < 0: Two complex conjugate roots a ± bi (parabola never touches x-axis)
Parabola Vertex: h = -b / (2a),  k = c - (b2 / 4a)

How to Solve Quadratic Equations by Hand (Step-by-Step Guide)

  1. Write in Standard Form: Rearrange terms so all terms are on one side set equal to zero: ax2 + bx + c = 0.
  2. Identify Coefficients: Write down numerical values for a (coefficient of x2), b (coefficient of x), and c (constant number).
  3. Calculate the Discriminant: Evaluate Δ = b2 - 4ac to determine how many real solutions exist.
  4. Apply the Quadratic Formula: Substitute a, b, and Δ into x = (-b ± √Δ) / (2a).
  5. Split into Plus and Minus Cases: Calculate x1 using "+ √Δ" and calculate x2 using "- √Δ".
  6. Simplify Answers: Express roots as exact simplified fractions/radicals and check solutions by plugging them back into the original equation.

3 In-Depth Worked Examples with Complete Solutions

Example 1: Two Distinct Real Roots (Factoring Verification)

Problem: Solve 2x2 + 5x - 3 = 0.

Step-by-Step Solution:

  • Coefficients: a = 2, b = 5, c = -3.
  • Calculate discriminant: Δ = (5)2 - 4(2)(-3) = 25 - (-24) = 25 + 24 = 49.
  • Since Δ = 49 > 0, there are two rational real roots. Note that √49 = 7.
  • Apply formula: x = [-5 ± 7] / (2 · 2) = [-5 ± 7] / 4.
  • Root 1 (+): x1 = (-5 + 7) / 4 = 2 / 4 = 1/2 = 0.5.
  • Root 2 (-): x2 = (-5 - 7) / 4 = -12 / 4 = -3.

Final Roots: x = 1/2, x = -3

Example 2: Complex Conjugate Roots (Δ < 0)

Problem: Solve x2 - 4x + 13 = 0.

Step-by-Step Solution:

  • Coefficients: a = 1, b = -4, c = 13.
  • Calculate discriminant: Δ = (-4)2 - 4(1)(13) = 16 - 52 = -36.
  • Since Δ < 0, the roots are imaginary: √(-36) = √(36) · √(-1) = 6i.
  • Apply formula: x = [-(-4) ± 6i] / (2 · 1) = [4 ± 6i] / 2.
  • Divide both real and imaginary terms by 2: x = 2 ± 3i.

Complex Roots: x = 2 + 3i, x = 2 - 3i

Example 3: Real-World Ballistics & Kinematics

Scenario: A flare is launched from a 40-foot platform with initial upward velocity of 96 ft/s. Its height in feet after t seconds is given by h(t) = -16t2 + 96t + 40. At what exact second does the flare crash into the ground (h = 0)?

Step-by-Step Solution:

  • Set equation: -16t2 + 96t + 40 = 0.
  • Simplify by dividing all terms by -8: 2t2 - 12t - 5 = 0 (a = 2, b = -12, c = -5).
  • Discriminant: Δ = (-12)2 - 4(2)(-5) = 144 + 40 = 184. Note √184 = √(4 × 46) = 2√46 ≈ 13.565.
  • Apply formula: t = [12 ± 13.565] / 4.
  • t1 = (12 + 13.565) / 4 = 25.565 / 4 ≈ 6.39 seconds.
  • t2 = (12 - 13.565) / 4 = -1.565 / 4 ≈ -0.39 s (discard negative time).

Landing Time: The flare hits the ground at t ≈ 6.39 seconds.

Common Quadratic Mistakes to Avoid

  • Sign Error in -b: If b = -6, then -b = -(-6) = +6. Students often forget to flip the sign of b.
  • Squaring Negative b Incorrectly: In the discriminant, (-4)2 is always positive +16, never -16.
  • Dividing Only One Term by 2a: The entire numerator [-b ± √Δ] must be divided by 2a. Dividing only the radical leads to completely invalid roots.
  • Not Setting to Zero First: If given x2 + 5x = 14, you MUST subtract 14 to make it x2 + 5x - 14 = 0 before identifying coefficients.

Real-World & Industry Applications

Quadratic equations are ubiquitous across STEM and business. In telecommunications, parabolic satellite dishes use quadratic curves because parallel radio waves reflect off the dish directly into a single focal point receiver. In business finance, revenue curves are quadratic (price times demand); finding the parabola vertex reveals the maximum profit price point. In automotive safety, braking distance increases quadratically with vehicle speed (doubling speed quadruples stopping distance).

Frequently Asked Questions About Quadratic Equations & CalcSolver

What does a discriminant of zero mean on a graph?
When Δ = 0, the parabola touches the horizontal x-axis at exactly one point (its vertex). The equation has one repeated rational real root.
How do I find the vertex of a parabola?
The x-coordinate of the vertex is located at h = -b / (2a). To find the y-coordinate k, substitute h back into the quadratic equation: k = a(h)2 + b(h) + c.
Can CalcSolver solve quadratic equations with complex numbers?
Yes. When the discriminant is negative (Δ < 0), CalcSolver automatically calculates and displays both imaginary conjugate roots in standard a ± bi form.
Is CalcSolver's quadratic solver free for high school students?
CalcSolver is 100% free with no paywalls, accounts, or subscriptions. It functions seamlessly on Chromebooks, laptops, and smartphones.