± surdAlgebra · Quadratics
Quadratic Formula Calculator with four methods
Enter a, b and c — or the whole equation. Solve it with the quadratic formula (Sridharacharya's formula in India), by factorising, by completing the square or from the graph.
- Free steps
- Graph
- Checked answers
Try it
Lift the parabola. Watch the roots.
When the discriminant hits zero the two roots merge; below zero they leave the real line as a complex pair.
Equation
x^2 - 2x - 2.00 = 0Vertex form
y = (x - 1)^2 - 3.00Roots
x = -0.73, 2.73D > 0: two real roots — the parabola crosses the x-axis twice.
Drag the parabola up or down, use the slider, or focus the graph and press ↑ ↓ (Shift for bigger steps).
Why this quadratic formula calculator is different
01
Three methods, one answer
The quadratic formula is shown in full, then the same roots by splitting the middle term (when the roots are rational) and by completing the square — so you can follow whichever method your class uses.
02
Exact surds and complex roots
√(40) becomes 2√(10), common factors cancel, and negative discriminants give roots like -2 ± i. Decimals are shown alongside, never instead.
03
Discriminant questions too
Type "find k if kx^2 + 4x + 1 = 0 has equal roots" and get D in terms of k, the value for equal roots and the ranges for real or no real roots — each re-checked from your equation.
What is a quadratic formula calculator?
A quadratic formula calculator finds the roots (also called the zeros or solutions) of any quadratic equation ax^2 + bx + c = 0 with a ≠ 0, and shows the working. The quadratic formula gives both roots directly: x = (-b ± √(b^2 - 4ac))/(2a). In Indian textbooks (CBSE and ICSE Class 10) it is called the Sridharacharya formula, after the mathematician Śrīdhara.
The expression under the root, D = b^2 - 4ac, is the discriminant. It tells you the nature of the roots before you finish: two distinct real roots when D > 0 (rational if D is a perfect square), one repeated root when D = 0, and two complex conjugate roots when D < 0.
The formula comes from completing the square, so it always works — even when the quadratic does not factorise. The roots are also the x-intercepts of the parabola y = ax^2 + bx + c. It appears in GCSE and IGCSE maths, Algebra 1 and 2, and IB and A-level courses.
How to use the quadratic formula calculator
- 1Type the equation, for example `2x^2 - 7x + 3 = 0`, or the coefficients as `a=2, b=-7, c=3` (any order). Equations such as `3x^2 - 5x = 2` are rearranged into standard form for you.
- 2Read the roots on the answer card, with exact values and decimals. Complex roots are written as p ± qi.
- 3Open the steps: identify a, b, c, work out the discriminant, substitute into the formula and simplify the surd. Methods 2 and 3 show factorisation and completing the square.
- 4Check the facts row for the discriminant, sum and product of roots, vertex and axis of symmetry.
- 5Open the Graph tab to see the parabola, its roots on the x-axis and its vertex, then try a similar practice problem.
Ways to solve a quadratic equation
Every quadratic can be solved with the formula; the other methods are faster when they apply.
Quadratic (Sridharacharya) formula
Works for every quadratic. Put the equation in standard form first so the right-hand side is 0.
The discriminant and the nature of the roots
D > 0: two real roots; D = 0: one repeated root; D < 0: no real roots (two complex roots). If D is a perfect square the roots are rational.
Finding k for equal or real roots
Write D in terms of the unknown constant, then solve D = 0, D > 0 or D < 0 — remembering that a must not be 0.
Factorisation (splitting the middle term)
Find two numbers whose product is ac and sum is b, split bx into px + qx, group in pairs and use the zero product property.
Completing the square and vertex form
Divide by a, add the square of half the x-coefficient to both sides and take square roots. The same working gives the vertex form y = a(x - h)^2 + k and the turning point (h, k).
Sum and product of the roots
Quick checks on your answers.
Quadratic formula calculator worked example: 2x² − 4x − 3 = 0
- 01
Identify the coefficients.
a = 2, b = -4, c = -3 - 02
Work out the discriminant. D = 40 > 0 but is not a perfect square, so there are two distinct irrational roots.
D = (-4)^2 - 4 × 2 × (-3) = 16 + 24 = 40 - 03
Substitute into the quadratic formula.
x = (-(-4) ± √(40))/(2 × 2) = (4 ± √(40))/(4) - 04
Simplify the surd: √(40) = √(4 × 10) = 2√(10).
x = (4 ± 2√(10))/(4) - 05
Divide the top and bottom by 2.
x = (2 ± √(10))/(2) ≈ 2.581 or -0.581 - 06
Check with the sum and product of roots: (2 + √(10))/(2) + (2 - √(10))/(2) = 2 = -b/a and the product is (4 - 10)/(4) = -3/2 = c/a.
Common mistakes with the quadratic formula
Losing the sign of b
The formula starts with -b. When b = -4, -b = +4. Writing -4 gives the wrong roots.
Not using standard form
For 3x^2 - 5x = 2, c is -2, not 2. Move every term to one side first so the equation equals 0.
Dividing only part of the top by 2a
In (4 ± 2√(10))/(4), divide both 4 and 2√(10) by the common factor: (2 ± √(10))/(2), not 1 ± 2√(10).
Sign errors in the discriminant
With a negative c, -4ac becomes positive: for a = 2, c = -3 it is -4(2)(-3) = +24. Use brackets when substituting negatives.
Quadratic formula calculator FAQ
Is the quadratic formula calculator free?+
Yes. Every quadratic — with all the steps, the three methods, the discriminant, the vertex form, the graph and practice problems — is free with no sign-up. The full working is shown on the page, so you can compare it line by line with your own and see exactly where an answer went wrong.
What is the Sridharacharya formula?+
It is the name used in Indian textbooks for the quadratic formula x = rac-b pm sqrtb^2 - 4ac2a, after the mathematician Śrīdhara, who described solving quadratics by completing the square. It is exactly the same formula taught as "the quadratic formula" in GCSE, Algebra 1 and IB courses.
What does the discriminant tell me about the roots?+
D = b^2 - 4ac gives the nature of the roots before you solve: two distinct real roots if D > 0 (rational when D is a perfect square), one repeated root if D = 0, and two complex conjugate roots if D < 0. On the graph that means the parabola crosses, touches or misses the x-axis.
How do I find the zeros of a quadratic function?+
The zeros of f(x) = ax^2 + bx + c are the roots of ax^2 + bx + c = 0 — the x-intercepts of its graph. Type the expression on its own, for example `2x^2 - x - 6`, or write `find the zeros of 2x^2 - x - 6`; the calculator sets it equal to zero and solves it.
Can it find complex roots?+
Yes. When D < 0 the roots are written exactly as p pm qi with any surd simplified, for example x^2 + 2x + 3 = 0 gives x = -1 pm sqrt2,i. Both roots are checked by substituting them back with complex arithmetic, and the graph shows why there are no real roots.
Can I enter just a, b and c?+
Yes — type `a=2, b=-7, c=3` in any order, `a = 2 and b = 3` (a missing c is 0) or simply `2, -7, 3`. You can also type a full equation such as `3x^2 - 5x = 2` or `x(x + 3) = 10`, and it is rearranged into standard form before solving.
How do I find k so that a quadratic has equal roots?+
Type the question, for example `find k if x^2 + kx + 9 = 0 has equal roots`. The calculator writes D = k^2 - 36, solves D = 0 to get k = pm 6, and also gives the values of k for two distinct real roots and for no real roots, checking each one from your original equation.
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