Quadratic Formula Calculator

How to Use the Quadratic Formula Calculator

1. Write your equation in standard form — ax² + bx + c = 0. If needed, move all terms to one side first.
2. Enter coefficient a — the number multiplying x². Must not be zero.
3. Enter coefficient b — the number multiplying x. Can be any real number including zero.
4. Enter constant c — the constant term (no x). Can be any real number including zero.
5. Read results instantly — both roots appear along with the discriminant, vertex, and axis of symmetry. All working steps are shown in the secondary panel.

Example: For x² − 5x + 6 = 0, enter a=1, b=−5, c=6. The calculator returns x₁=3 and x₂=2.

The Discriminant: Key to Understanding the Roots

The discriminant is the expression inside the square root in the quadratic formula: Δ = b² − 4ac. It determines the nature of the roots before you even compute them, which is why it is called the discriminant — it discriminates between the three possible cases.

Case 1: Discriminant > 0 — Two Distinct Real Roots

When b² − 4ac > 0, the square root exists and is a positive number. The ± in the formula produces two different real numbers. The parabola crosses the x-axis at two distinct points. If the discriminant is also a perfect square (e.g., 1, 4, 9, 25), the roots are rational numbers; otherwise they are irrational surds.

Case 2: Discriminant = 0 — One Repeated Real Root

When b² − 4ac = 0, the square root is zero, so the ± term vanishes. Both formulas give the same result: x = −b / (2a). This is a repeated (double) root. The parabola touches the x-axis at exactly one point — the vertex — and does not cross it.

Case 3: Discriminant < 0 — Two Complex Roots

When b² − 4ac < 0, the square root of a negative number is required. In the real number system this has no solution, but in the complex number system it gives two conjugate complex roots: x = (−b ± i√|Δ|) / (2a), where i = √(−1). The parabola lies entirely above or below the x-axis with no real intersections. Complex roots always come in conjugate pairs (a + bi and a − bi).

Worked Example: x² − 5x + 6 = 0

1. Identify: a=1, b=−5, c=6
2. Discriminant: Δ = (−5)² − 4(1)(6) = 25 − 24 = 1 > 0 → two distinct real roots
3. x = (−(−5) ± √1) / (2×1) = (5 ± 1) / 2
4. x₁ = 6/2 = 3, x₂ = 4/2 = 2
5. Verify: (3)² − 5(3) + 6 = 9 − 15 + 6 = 0 ✓ and (2)² − 5(2) + 6 = 4 − 10 + 6 = 0 ✓

Vertex, Axis of Symmetry, and Graphing Parabolas

The quadratic equation y = ax² + bx + c describes a parabola. Understanding its key features allows you to sketch or analyse the graph without plotting many points.

The Vertex

The vertex is the turning point of the parabola. Its x-coordinate is x_v = −b / (2a) — this is the axis of symmetry. The y-coordinate is found by substituting x_v back into the equation: y_v = c − b² / (4a) (or equivalently, f(x_v)). If a > 0, the vertex is the global minimum (parabola opens upward). If a < 0, the vertex is the global maximum (parabola opens downward).

Axis of Symmetry

The axis of symmetry is the vertical line x = −b / (2a). The parabola is perfectly symmetric about this line. Both roots x₁ and x₂ (when real) are equidistant from it: their midpoint is (x₁ + x₂) / 2 = −b / (2a), which also equals the x-coordinate of the vertex.

Direction and Width

The sign of a determines direction: a > 0 opens upward (U-shape), a < 0 opens downward (∩-shape). The magnitude |a| determines width: larger |a| means a narrower parabola; smaller |a| means wider. When a = 1, the parabola has the standard width used as a reference.

Vertex Form

Any quadratic can be written in vertex form: y = a(x − h)² + k, where (h, k) is the vertex. This form makes it immediately clear where the vertex is and in which direction the parabola opens, without requiring any calculations.

Applications of Quadratic Equations

Quadratic equations model many real-world phenomena: the trajectory of a projectile (height vs time under gravity), the stopping distance of a vehicle, the area of a rectangle given its perimeter, profit maximisation in economics, and the shape of reflective dishes in satellite antennas and car headlights (which use parabolic reflectors to concentrate signals or light).

Worked Examples: Solving Quadratic Equations

How do I solve 2x² + 3x − 2 = 0?

x = 0.5 and x = −2. a=2, b=3, c=−2. Δ = 9 − 4(2)(−2) = 9 + 16 = 25. x = (−3 ± 5) / 4. x₁ = 2/4 = 0.5, x₂ = −8/4 = −2.

How do I solve x² + 4x + 4 = 0?

x = −2 (repeated root). Δ = 16 − 16 = 0. x = −4 / 2 = −2. The parabola touches the x-axis at one point.

How do I solve x² + x + 1 = 0?

Two complex roots: x = −0.5 ± i(√3/2). Δ = 1 − 4 = −3 < 0. x = (−1 ± i√3) / 2.

What is the vertex of y = x² − 6x + 5?

Vertex: (3, −4). x_v = −(−6) / (2×1) = 3. y_v = (3)² − 6(3) + 5 = 9 − 18 + 5 = −4. The parabola has a minimum at (3, −4).

How do I find where y = −x² + 4x − 3 crosses the x-axis?

x = 1 and x = 3. a=−1, b=4, c=−3. Δ = 16 − 12 = 4. x = (−4 ± 2) / (−2). x₁ = (−4+2)/(−2) = 1, x₂ = (−4−2)/(−2) = 3.

How do I convert ax²+bx+c=0 to vertex form?

Complete the square or use h = −b/2a and k = c − b²/4a. For x²−4x+1: h=2, k=1−4=−3. Vertex form: (x−2)² − 3.

People Also Search For

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

The quadratic formula is x = (−b ± √(b²−4ac)) / 2a and solves ax²+bx+c=0. Identify a, b, and c from your equation, substitute into the formula, compute the discriminant, then evaluate the two ± versions for x₁ and x₂.

How to solve a quadratic equation by factoring

Find two numbers that multiply to a×c and add to b, then factor. For x²−5x+6: find numbers multiplying to 6 and adding to −5: those are −2 and −3. Factor: (x−2)(x−3)=0 → x=2 or x=3. Factoring is faster when integer roots exist; the quadratic formula always works.

What is completing the square?

Completing the square rewrites ax²+bx+c as a(x+h)²+k by adding and subtracting (b/2a)². It reveals the vertex and is the derivation of the quadratic formula itself. For x²+6x+5: (x+3)²−9+5 = (x+3)²−4 → x+3=±2 → x=−1 or x=−5.

How to use the quadratic formula for complex roots

When b²−4ac < 0, replace √(negative) with i×√(|negative|). For x²+2x+5: Δ=4−20=−16. x=(−2±√(−16))/2=(−2±4i)/2=−1±2i. The roots are −1+2i and −1−2i.

What is the sum and product of quadratic roots?

Sum of roots = −b/a and Product of roots = c/a. These Vieta's formulas let you check your answers without substituting back. For x²−5x+6=0: sum = 5/1 = 5 (3+2=5 ✓), product = 6/1 = 6 (3×2=6 ✓).

Frequently Asked Questions