Quadratic Equation Calculator with Steps

Q: What is the quadratic formula?

The quadratic formula is x = (−b ± √(b² − 4ac)) ÷ 2a. It solves any quadratic equation ax² + bx + c = 0 where a is not zero. The expression b² − 4ac is called the discriminant.

Q: How do I solve a quadratic equation with this calculator?

Put the equation in the form ax² + bx + c = 0, enter a, b and c (including minus signs) and read the roots. The calculator shows every step of the quadratic formula along with the discriminant and vertex.

Q: Can the quadratic equation calculator find complex roots?

Yes. When the discriminant is negative, the calculator gives both complex roots in the form p ± qi, showing the real and imaginary parts separately.

Q: What does the discriminant tell you?

If b² − 4ac is positive there are two different real roots, if it is zero there is one repeated root, and if it is negative the roots are complex conjugates and the parabola does not touch the x-axis.

Q: How do I find the vertex of a quadratic equation?

The x-coordinate of the vertex is h = −b ÷ 2a. Substitute h into the equation to get k, or use k = c − b² ÷ 4a. The vertex form is y = a(x − h)² + k.

Q: How do you calculate a quadratic equation in a scientific calculator?

On Casio models, open the Equation (EQN) mode, choose the degree-2 polynomial option, enter a, b and c and press equals to see both roots. The online calculator here gives the same roots with steps.

Q: What if a = 0?

Then the equation is not quadratic but linear: bx + c = 0, so x = −c ÷ b. The calculator detects this and solves the linear equation.

Q: What is a quadratic equation in class 10 maths?

In class 10 a quadratic equation is defined as ax² + bx + c = 0 with a ≠ 0. Students solve it by factorisation, completing the square and the quadratic formula, and use the discriminant to find the nature of roots.

Q: What is the name of the quadratic formula in class 10?

It is the quadratic formula, also called Sridharacharya's formula in NCERT class 10: x = (−b ± √(b² − 4ac)) ÷ 2a.

Q: Can two points define a quadratic equation?

Not on their own. You need three points, or two points where one is the vertex.

What is a quadratic equation calculator?

A quadratic equation calculator solves any equation of the form ax² + bx + c = 0 in one click. Type the three coefficients and this quadratic equation calculator with steps returns both roots, the discriminant, the vertex, the axis of symmetry and the factored form. It handles real roots, repeated roots and complex roots with imaginary numbers, so it works as a full online quadratic equation solver for homework, class 10 board revision, JEE practice or checking answers from a scientific calculator.

What is a quadratic equation?

A quadratic equation is a polynomial equation of degree two: the highest power of the variable x is 2. Its standard form is ax² + bx + c = 0, where a, b and c are real numbers and a ≠ 0. If a were 0, the x² term would disappear and you would have a linear equation instead. Examples: x² − 5x + 6 = 0, 2x² + 4x − 6 = 0 and 3x² − 12 = 0 (here b = 0). The solutions of the equation are called its roots or zeros, because they are the x-values where the graph y = ax² + bx + c crosses the x-axis.

How to use this quadratic equation solver

Rewrite your equation in standard form so that everything is on one side and 0 is on the other. For example, 2x² = 3x + 5 becomes 2x² − 3x − 5 = 0.
Enter a, the coefficient of x². Enter b, the coefficient of x, and c, the constant. Use 0 for a missing term and keep the minus signs.
Read the roots at the top. Below them you get the discriminant, the vertex and vertex form, the axis of symmetry and a full quadratic equation calculator with solution steps you can copy into your notebook.

If you set a = 0 the tool tells you the equation is linear and solves bx + c = 0 instead, so you never get a confusing “math error”.

The quadratic formula

Every quadratic equation can be solved with the quadratic formula, the method this solver uses:

x = (−b ± √(b² − 4ac)) ÷ 2a

a = coefficient of x² (not zero)
b = coefficient of x
c = constant term
b² − 4ac = the discriminant, written D or Δ

The formula comes from completing the square on ax² + bx + c = 0, which is why it is sometimes called the Sridharacharya formula in Indian textbooks, after the mathematician who described it. The ± sign gives the two roots: one with plus and one with minus.

Discriminant: nature of the roots

Before you solve, the discriminant tells you what kind of roots to expect. This is part of the class 10 syllabus (CBSE Chapter 4, Quadratic Equations).

Discriminant D = b² − 4ac	Nature of roots	Graph
D > 0 (perfect square)	Two distinct real, rational roots	Crosses the x-axis twice
D > 0 (not a perfect square)	Two distinct real, irrational roots	Crosses the x-axis twice
D = 0	One repeated real root, x = −b/2a	Touches the x-axis at the vertex
D < 0	Two complex conjugate roots p ± qi	Does not meet the x-axis

Worked example: solve 2x² + 4x − 6 = 0

Here a = 2, b = 4 and c = −6.

Discriminant: D = 4² − 4 × 2 × (−6) = 16 + 48 = 64.
√D = 8, a perfect square, so both roots are rational.
x = (−4 ± 8) ÷ 4, giving x₁ = 4 ÷ 4 = 1 and x₂ = −12 ÷ 4 = −3.
Vertex: h = −4 ÷ 4 = −1 and k = 2(−1)² + 4(−1) − 6 = −8, so the vertex is (−1, −8).
Factored form: 2(x − 1)(x + 3). Check: 2(x² + 2x − 3) = 2x² + 4x − 6.

Example with imaginary numbers: x² + 2x + 5 = 0

D = 4 − 20 = −16, which is negative, so there are no real roots. √(−16) = 4i, and x = (−2 ± 4i) ÷ 2 = −1 ± 2i. Enter a = 1, b = 2, c = 5 above and the quadratic equation calculator with imaginary numbers shows the same complex roots, the real part and the imaginary part.

Vertex, vertex form and graph

The graph of y = ax² + bx + c is a parabola. It opens upward when a > 0 and downward when a < 0. Its turning point, the vertex, sits at h = −b/2a, and k is the value of the function there. The calculator also rewrites your equation in vertex form, y = a(x − h)² + k, which is handy for sketching a graph by hand: plot the vertex, draw the axis of symmetry x = h, mark the y-intercept (0, c) and its mirror image, then join the points with a smooth curve. The roots are the points where the parabola cuts the x-axis, so the quadratic equation calculator vertex and point output doubles as a quick graph checklist.

Other ways to solve quadratic equations

By factoring

Solving a quadratic equation by factoring (factorisation) works when the roots are rational. For x² − 5x + 6 = 0, find two numbers that multiply to 6 and add to −5: −2 and −3. So (x − 2)(x − 3) = 0 and, by the zero product property, x = 2 or x = 3. For ax² + bx + c with a ≠ 1, split the middle term using two numbers whose product is ac. This calculator's factored form, a(x − x₁)(x − x₂), lets you check your factorisation.

By completing the square

Divide by a, move c across, then add (b/2a)² to both sides so the left side becomes a perfect square. For x² + 6x + 5 = 0: x² + 6x = −5, x² + 6x + 9 = 4, (x + 3)² = 4, so x + 3 = ±2 and x = −1 or −5. Completing the square is exactly how the quadratic formula is derived.

By extracting square roots

When b = 0, isolate x² and take square roots. For 3x² − 12 = 0: x² = 4, so x = ±2. Our square root calculator helps when the number under the root is not a perfect square.

How to solve a quadratic equation in a scientific calculator

Many students ask how to calculate a quadratic equation using a calculator such as the Casio fx-991EX ClassWiz or fx-991ES Plus. On most Casio models press MODE (or MENU), choose EQN (Equation), then the option for ax² + bx + c = 0, type a, b and c pressing = after each, and press = again to see x₁ and x₂. Newer ClassWiz models also show the vertex. On a TI-84 you can use the PlySmlt2 app or graph the function and use the zero feature. If your calculator shows “Math ERROR”, check that a is not zero. When you do not have such a calculator mode, this free online tool gives the same answer with every step shown, which is more useful for non-calculator exam questions where working earns marks.

Quadratic equation vs quadratic function

The difference between a quadratic equation and a quadratic function is simple: the function f(x) = ax² + bx + c describes a whole parabola and gives a y-value for every x, while the equation ax² + bx + c = 0 asks for the particular x-values where that function equals zero. Solving the equation finds the zeros of the function. Quadratic functions model projectile motion, area problems and profit curves; for example, the height of a ball thrown upward is a quadratic in time.

Tips to avoid mistakes

Always bring the equation to standard form before reading a, b and c.
Keep signs: in x² − 3x + 2 = 0, b is −3, not 3.
Square b before applying the sign: (−3)² = 9.
With large numbers, divide every term by a common factor first; the roots do not change.
Check a root by substituting it back into the equation. Our scientific calculator is useful for that check, and the exponent calculator for higher powers.

Quadratic equation calculator: solve for x step by step

To use the quadratic equation calculator to solve for x, first rearrange your equation so one side is zero. For 3x² = 5x + 2, subtract to get 3x² − 5x − 2 = 0, then input a = 3, b = −5, c = −2. The quadratic equation solver for x returns x = 2 and x = −1/3. These solutions are also the zeros of y = 3x² − 5x − 2, so this doubles as a quadratic equation zeros calculator.

Quadratic equation input: calculator tips for a, b and c

This quadratic equation solver takes a, b and c in the standard order ax² + bx + c = 0. Enter the sign with each coefficient: x² − 4 = 0 has b = 0 and c = −4.
If the x term is missing, b is 0; if the constant is missing, c is 0.
a must not be 0, or the equation is linear, not quadratic.

Quadratic equation solver formula: name and class 10 method

The quadratic equation solver formula is x = (−b ± √(b² − 4ac)) ÷ 2a. Its name is simply the quadratic formula; in Indian class 10 NCERT books it is also called Sridharacharya's formula, after the mathematician Sridhara. The quadratic formula x calculator above applies it to your a, b and c.

Quadratic equation calculator by factoring

Factoring works when the roots are whole numbers or simple fractions. For x² − 3x + 2 = 0 (the default), find two numbers that multiply to 2 and add to −3: −1 and −2. So (x − 1)(x − 2) = 0 and x = 1 or 2. For a quadratic equation calculator factoring check, the roots r₁ and r₂ from the calculator give the factors a(x − r₁)(x − r₂).

Quadratic equation calculator by extracting square roots

If there is no x term, isolate x² and take square roots. 2x² − 50 = 0 → x² = 25 → x = ±5. Enter a = 2, b = 0, c = −50 to confirm.

Quadratic equation calculator with two points

A parabola y = ax² + bx + c has three unknowns, so two points alone are not enough unless you know one more fact. If one point is the vertex (h, k), use y = a(x − h)² + k and the second point to find a. Vertex (1, 2) and point (3, 10): 10 = a × 4 + 2, so a = 2 and y = 2(x − 1)² + 2 = 2x² − 4x + 4.

Quadratic linear equation calculator: solving a system

To solve a quadratic and a linear equation together, substitute the line into the quadratic. For y = x² and y = x + 2: x² − x − 2 = 0, so enter a = 1, b = −1, c = −2. The roots are x = 2 and x = −1, giving the points (2, 4) and (−1, 1).

Quadratic equation calculator with variables

If the coefficients are letters, apply the formula symbolically. For x² − (p + q)x + pq = 0, the roots are p and q. For kx² + 4x + 1 = 0 to have equal roots, set b² − 4ac = 0: 16 − 4k = 0, so k = 4.

Quadratic equation on TI-84 calculator and calculator mode

For a quadratic equation on a TI-84 calculator, use the PlySmlt2 app (APPS → PlySmlt2 → Polynomial Root Finder, degree 2), or graph Y1 = ax² + bx + c and use 2nd → CALC → zero. On a Casio fx-991ES, the quadratic equation calculator mode is MODE 5 (EQN) then 3.

Quadratic equation non calculator method

In a non calculator paper, factor first, then complete the square, and use the formula only when the discriminant is a perfect square or the answer can stay in surd form. For x² + 6x + 4 = 0: (x + 3)² = 5, so x = −3 ± √5.

Best quadratic equation questions to practise

These quadratic formula calculator questions also work as quadratic formula non calculator questions:

x² − 5x + 6 = 0 → x = 2, 3
2x² + 3x − 2 = 0 → x = 1/2, −2
x² − 4x + 4 = 0 → x = 2 (repeated)
x² + 4x + 13 = 0 → x = −2 ± 3i
A ball's height is h = −5t² + 20t. When does it land? −5t² + 20t = 0 → t = 4 s

Quadratic equation calculator technique for checking answers

Check any answer with the sum and product of roots: r₁ + r₂ = −b/a and r₁ × r₂ = c/a. For 2x² + 3x − 2 = 0, 1/2 + (−2) = −3/2 and 1/2 × (−2) = −1. Both match.

Quadratic equation solver in Python and Java

A quadratic equation solver in Python can use cmath so complex roots work: d = cmath.sqrt(b*b - 4*a*c); x1 = (-b + d) / (2*a); x2 = (-b - d) / (2*a). A quadratic equation solver in Java uses Math.sqrt when the discriminant is non-negative and prints the real and imaginary parts separately when it is negative. No quadratic equation solver app is needed for everyday use: this page works on any phone.

Frequently asked questions

What is the quadratic formula?