What is the quadratic formula and how do I use it?

The quadratic formula solves any equation of the form ax² + bx + c = 0. The formula is x = (−b ± √(b² − 4ac)) / 2a. The ± symbol means you calculate two values: one using addition and one using subtraction, giving you both roots. Enter your values of a, b, and c into this calculator and it applies the formula automatically. Step-by-step for x² − 5x + 6 = 0 (a=1, b=−5, c=6): • Step 1: discriminant = (−5)² − 4(1)(6) = 25 − 24 = 1 • Step 2: x = (−(−5) ± √1) / (2 × 1) = (5 ± 1) / 2 • Step 3: x₁ = (5 + 1) / 2 = 3, x₂ = (5 − 1) / 2 = 2 • Answer: x = 3 or x = 2 • Verify: (3)² − 5(3) + 6 = 9 − 15 + 6 = 0 ✓

What is the discriminant and what does it tell me?

The discriminant is the expression inside the square root: D = b² − 4ac. It tells you how many real solutions the equation has before you do any further calculation. Discriminant interpretation: • D > 0: two distinct real roots — the parabola crosses the x-axis at two points • D = 0: exactly one real root (a repeated/double root) — the parabola just touches the x-axis at its vertex • D < 0: no real roots, two complex conjugate roots — the parabola does not cross the x-axis at all • D is a perfect square and a, b, c are integers: roots are rational numbers • D is positive but not a perfect square: roots are irrational (contain a square root) • The sign of D alone tells you the entire geometric relationship between the parabola and the x-axis

What are complex roots and what do they mean?

When the discriminant is negative (D 0) or below (if a < 0) the x-axis. Example — x² + x + 1 = 0 (D = 1 − 4 = −3): • x = (−1 ± √(−3)) / 2 = −0.5 ± (√3/2)i • x₁ = −0.5 + 0.866i, x₂ = −0.5 − 0.866i • These cannot be plotted on the real number line • The parabola sits entirely above the x-axis with vertex at (−0.5, 0.75) • Complex roots are common in physics (damped oscillations), engineering (control systems), and signal processing

What is the vertex of a parabola and how is it calculated?

The vertex is the turning point of the parabola — the minimum point if a > 0 (parabola opens upward) or the maximum point if a 0, the vertex is below the x-axis (for a>0) or above (for a 0) — no real x-intercepts • The vertex gives the minimum/maximum value of the quadratic function • In physics: vertex = peak height of a projectile; in economics: vertex = maximum profit point

What is the relationship between roots, sum, and product (Vieta's formulas)?

For a quadratic ax² + bx + c = 0 with roots x₁ and x₂, Vieta's formulas state: Sum of roots = x₁ + x₂ = −b/a. Product of roots = x₁ × x₂ = c/a. These hold even for complex roots and can be used to verify your answer without substituting back into the equation. Vieta's formulas in practice: • x² − 5x + 6 = 0: sum = 5/1 = 5 (roots 3+2=5 ✓), product = 6/1 = 6 (roots 3×2=6 ✓) • 2x² − 4x − 6 = 0: sum = 4/2 = 2 (roots 3+(−1)=2 ✓), product = −6/2 = −3 (roots 3×(−1)=−3 ✓) • Useful for constructing a quadratic from its roots: if roots are 4 and −2, then sum=2, product=−8, equation is x² − 2x − 8 = 0 • Vieta's formulas generalise to polynomials of any degree

Quadratic Equation Solver — Roots, Vertex & Parabola

This Quadratic Equation Solver finds the roots of any quadratic equation ax² + bx + c = 0 using the quadratic formula. Enter the three coefficients a, b, and c — the solver instantly shows both roots (real or complex), the discriminant, vertex coordinates, axis of symmetry, y-intercept, sum and prod...

ax² + bx + c = 0

a (x² coefficient)

must not be 0

b (x coefficient)

can be 0

c (constant)

can be 0

x² − 5x + 6 = 0

TWO REAL ROOTS

ROOT 1 (x₁)

ROOT 2 (x₂)

DISCRIMINANT

VERTEX

(2.5, -0.25)

y-INTERCEPT

QUICK EXAMPLES

PARABOLA DIAGRAM

STEP-BY-STEP SOLUTION

Identify: a = 1, b = -5, c = 6

Discriminant: b² − 4ac = -5² − 4(1)(6) = 1

D > 0 → two distinct real roots

x = (−b ± √D) / 2a = (5 ± √1) / 2

x₁ = 3, x₂ = 2

PARABOLA PROPERTIES

Opens	Upward ↑
Vertex	(2.5, -0.25)
Axis of symmetry	x = 2.5
y-intercept	(0, 6)
Sum of roots	5
Product of roots	6

Created with❤️byeaglecalculator.com

HOW TO USE

1
Enter coefficient a (the x² term) — must not be zero. Enter b (the x term) — can be zero. Enter c (the constant term) — can be zero.
2
The equation display updates instantly to show the formatted equation. The solver applies the quadratic formula x = (−b ± √(b² − 4ac)) / 2a automatically.
3
The discriminant (b² − 4ac) tells you the root type before reading the roots: positive = two real roots, zero = one double root, negative = two complex roots.
4
Read the roots in the result box. Complex roots are shown as p ± qi. The parabola diagram shows the curve, vertex, axis of symmetry, and x-intercepts (if real roots exist).
5
Use the step-by-step panel on the right to follow the complete solution. Load quick examples to explore different equation types.

WORKED EXAMPLE

Example 1 (two real roots): x² − 5x + 6 = 0. D = 25 − 24 = 1. x = (5 ± 1)/2. x₁ = 3, x₂ = 2. Vertex: (2.5, −0.25). Example 2 (double root): x² + 2x + 1 = 0. D = 4 − 4 = 0. x = −1 (double root). Vertex: (−1, 0). Example 3 (complex): x² + x + 1 = 0. D = 1 − 4 = −3. x = −0.5 ± (√3/2)i ≈ −0.5 ± 0.866i.

REFERENCE FORMULAS

FORMULA REFERENCE

NAME	FORMULA	DESCRIPTION
Quadratic Formula	x = (−b ± √(b² − 4ac)) / 2a	Solves any quadratic — the universal method
Discriminant	D = b² − 4ac	D>0: two real roots, D=0: one root, D<0: complex roots
Vertex x	x_v = −b / (2a)	x-coordinate of the parabola turning point
Vertex y	y_v = ax_v² + bx_v + c	y-coordinate — min if a>0, max if a<0
Sum of roots	x₁ + x₂ = −b / a	Vieta formula — verify without substituting
Product of roots	x₁ × x₂ = c / a	Vieta formula — second verification check

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Last updated: April 28, 2026 · Formula verified by EagleCalculator team · Eagle-eyed accuracy for every calculation.