---
title: "Quadratic Equation Calculator | Method Calculator"
description: "Solve ax² + bx + c = 0 with the quadratic formula. Shows the discriminant, real or complex roots, the vertex and every step of the working."
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> Index of every page on this site: [llms.txt](https://methodcalculator.com/llms.txt)

# **Quadratic Equation Calculator**

Enter the coefficients a, b and c of ax² + bx + c = 0 to get the roots, the discriminant, the vertex and the factored form, with every step of the quadratic formula written out.

## **Result**

Roots

**x = 2 or x = 1**

<dl>

<dt>Root type</dt>
<dd>**Two distinct real roots**</dd>

<dt>Discriminant (b² − 4ac)</dt>
<dd>1</dd>

<dt>Vertex</dt>
<dd>(1.5, −0.25)</dd>

<dt>Axis of symmetry</dt>
<dd>x = 1.5</dd>

<dt>Parabola opens</dt>
<dd>Upward (vertex is the minimum)</dd>

<dt>Factored form</dt>
<dd>(x − 2)(x − 1) = 0</dd>

<dt>Sum of roots (−b ÷ a)</dt>
<dd>3</dd>

<dt>Product of roots (c ÷ a)</dt>
<dd>2</dd>

</dl>

### **Step-by-step solution **

1. Identify the coefficients: a = 1, b = −3, c = 2.
2. Compute the discriminant: D = b² − 4ac = (−3)² − 4 × 1 × 2 = 9 − 8 = 1.
3. D > 0, so the equation has two distinct real roots.
4. Apply the quadratic formula: x = (−b ± √D) ÷ 2a = (3 ± √1) ÷ 2 = (3 ± 1) ÷ 2.
5. x₁ = (3 + 1) ÷ 2 = 2 and x₂ = (3 − 1) ÷ 2 = 1.
6. Vertex: x = −b ÷ 2a = 1.5, y = c − b² ÷ 4a = −0.25, so the vertex is (1.5, −0.25).

## What is a quadratic equation?

A quadratic equation is any equation that can be written as **ax² + bx + c = 0**, where a, b and c are known numbers and a is not zero. The highest power of x is 2, which gives the equation its name (*quadratus* is Latin for square). Quadratics appear wherever one quantity depends on the square of another: the height of a thrown ball over time, braking distance against speed, the area of a garden with a fixed length of fence, or the profit of a product whose margin shrinks as volume grows. Solving the equation means finding the values of x, called the roots or solutions, that make the left-hand side equal to zero.

## The quadratic formula

Every quadratic equation, including those that do not factor neatly, is solved by:

**x = (−b ± √(b² − 4ac)) ÷ 2a**

The ± sign produces two values, one using plus and one using minus. The expression under the square root, b² − 4ac, is called the discriminant and decides what kind of roots you get.

**Example:** solve x² − 3x + 2 = 0, so a = 1, b = −3 and c = 2.

1. Discriminant: D = (−3)² − 4 × 1 × 2 = 9 − 8 = 1.
2. Square root: √1 = 1.
3. Formula: x = (3 ± 1) ÷ 2, so x₁ = 4 ÷ 2 = **2** and x₂ = 2 ÷ 2 = **1**.
4. Check: 2² − 3 × 2 + 2 = 4 − 6 + 2 = 0 and 1² − 3 × 1 + 2 = 0. Both work.

Because the roots are 2 and 1, the quadratic factors as (x − 2)(x − 1) = 0. The sum of the roots is 3, which equals −b ÷ a, and the product is 2, which equals c ÷ a. These two relationships (Vieta's formulas) are a quick way to check any answer.

## What the discriminant tells you

| Discriminant D = b² − 4ac | Roots | Graph |
| --- | --- | --- |
| D > 0 | Two distinct real roots | Parabola crosses the x-axis twice |
| D = 0 | One repeated real root, x = −b ÷ 2a | Parabola touches the x-axis at its vertex |
| D < 0 | Two complex conjugate roots | Parabola never reaches the x-axis |
| D is a perfect square (integer coefficients) | Rational roots; the quadratic factors over the integers | Crosses the x-axis at rational points |

## Completing the square

The quadratic formula is what you get by completing the square on the general equation, and the method is worth knowing on its own. For x² − 3x + 2 = 0: move the constant across (x² − 3x = −2), add the square of half the x-coefficient to both sides ((3 ÷ 2)² = 2.25, giving x² − 3x + 2.25 = 0.25), rewrite the left side as a perfect square ((x − 1.5)² = 0.25), take the square root of both sides (x − 1.5 = ±0.5) and solve: x = 2 or x = 1. The intermediate form a(x − h)² + k is called vertex form, because (h, k) is the vertex of the parabola.

## What the graph shows

The graph of y = ax² + bx + c is a parabola. It opens upward when a is positive and downward when a is negative. The real roots are the points where the curve crosses the x-axis, and the constant c is where it crosses the y-axis. The vertex sits on the axis of symmetry x = −b ÷ 2a and has a y-coordinate of c − b² ÷ 4a. For the example, the vertex is (1.5, −0.25): the curve dips just below the axis between the two roots, which is why there are two of them. If you need the gradient of a straight line instead, see the [slope calculator](https://methodcalculator.com/slope-calculator).

## How to use this calculator

1. Rearrange your equation into standard form with zero on the right-hand side. 2x² = 5x − 3 becomes 2x² − 5x + 3 = 0.
2. Type a, b and c with their signs. Use 0 for a missing term: x² − 9 = 0 has a = 1, b = 0, c = −9.
3. Check the rendered equation under the inputs matches what you meant to solve.
4. Read the roots, the discriminant and the vertex, and follow the numbered steps to see the working.

## Common mistakes

- **Sign errors with b:** when b = −3, −b is +3, and b² is +9, never −9. Squaring a negative gives a positive.
- **Forgetting the ±:** a quadratic normally has two roots. Writing only the "+" version loses one of them.
- **Dividing only part of the top:** the whole of −b ± √D is divided by 2a, not just the square root.
- **Reading coefficients out of order:** in 5 + x² = 4x, a = 1, b = −4 and c = 5 once it is in standard form.
- **Rounding too early:** keep √D exact until the last step. The calculator shows up to 4 decimal places, so irrational roots are rounded.
- **Fractional coefficients:** clear them first with the [fraction calculator](https://methodcalculator.com/fraction-calculator) or multiply through by the common denominator; powers and roots are covered by the [exponent calculator](https://methodcalculator.com/exponent-calculator).

## **Frequently asked questions **

### **What happens if a is zero?**

Then there is no x² term and the equation is linear, not quadratic: bx + c = 0 has the single solution x = −c ÷ b. The calculator detects this and solves it directly instead of dividing by zero in the quadratic formula. If b is also zero the equation has either no solution (c ≠ 0) or every x as a solution (c = 0).

### **Why does the calculator show roots with an "i"?**

When the discriminant b² − 4ac is negative, the formula asks for the square root of a negative number. There is no real number with that property, so the roots are written with the imaginary unit i = √(−1). The two roots are complex conjugates, such as 1.5 + 1.3229i and 1.5 − 1.3229i, and the parabola never crosses the x-axis.

### **Can I enter fractions or decimals as coefficients?**

Decimals work directly: type 0.5 for ½ or 0.333333 for ⅓. For exact fractions, multiply the whole equation by the common denominator first, which does not change the roots. ½x² + ¼x − 3 = 0 becomes 2x² + x − 12 = 0 after multiplying by 4.

### **How can I tell if a quadratic can be factored?**

With integer coefficients, the quadratic factors over the integers exactly when the discriminant is a perfect square (0, 1, 4, 9, 16 …). Then the roots are rational and the factored form row shows the factors. If the discriminant is positive but not a perfect square, the roots are real but irrational, such as (1 + √5) ÷ 2.

### **What is the vertex used for?**

The vertex is the highest or lowest point of the parabola, so it answers maximum and minimum questions: the peak height of a projectile, the price that maximises revenue, or the size that minimises cost. Its x-coordinate is −b ÷ 2a and its y-coordinate is the value of the quadratic at that x.

## **Related calculators **

- [**Exponent Calculator**](https://methodcalculator.com/exponent-calculator)
- [**Slope Calculator**](https://methodcalculator.com/slope-calculator)
- [**Fraction Calculator**](https://methodcalculator.com/fraction-calculator)
- [**Percentage Calculator**](https://methodcalculator.com/percentage-calculator)

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