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Quadratic Equation With Complex Roots

Solve a quadratic with no real roots and get the complex pair, with the working.

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Enter
of x squared
of x
constant
Try:

Roots

x1=−1+2 ix_1 = -1 + 2\,ix1​=−1+2i
x2=−1−2 ix_2 = -1 - 2\,ix2​=−1−2i

No real roots. The complex roots are -1 + 2i and -1 - 2i.

Forms of the same parabola

Standard formy=x2+2x+5y = x^{2} + 2x + 5y=x2+2x+5
Factored formy=(x−(−1+2 i))(x−(−1−2 i))y = \left(x - \left(-1 + 2\,i\right)\right)\left(x - \left(-1 - 2\,i\right)\right)y=(x−(−1+2i))(x−(−1−2i))
Vertex formy=(x+1)2+4y = \left(x + 1\right)^2 + 4y=(x+1)2+4
Vertex(−1, 4)\left(-1,\ 4\right)(−1, 4)
Axis of symmetryx=−1x = -1x=−1
y-intercept(0, 5)\left(0,\ 5\right)(0, 5)
DiscriminantΔ=−16\Delta = -16Δ=−16

No real factors. This is the factorisation over the complex numbers. The parabola opens upward, so the vertex is a minimum.

Parabola

  • y
Table of values
Graph of the parabola with its roots and vertex marked: sampled values across the visible range
xy
-2.45.96
-2.125.2544
-1.844.7056
-1.564.3136
-1.284.0784
-14
-0.724.0784
-0.444.3136
-0.164.7056
0.125.2544
0.45.96

Quadratic formula, step by step

  1. 1Identify the coefficients
    ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0
    ↓
    x2+2x+5=0x^{2} + 2x + 5 = 0x2+2x+5=0

    a = 1, b = 2, c = 5

  2. 2Discriminant
    Δ=b2−4ac\Delta = b^2 - 4acΔ=b2−4ac
    ↓
    Δ=22−4(1)5=−16\Delta = 2^2 - 4(1)5 = -16Δ=22−4(1)5=−16

    Negative: no real roots, two complex roots.

  3. 3Quadratic formula
    x=−b±Δ2ax = \frac{-b \pm \sqrt{\Delta}}{2a}x=2a−b±Δ​​
    ↓
    x=−2±−162x = \frac{-2 \pm \sqrt{-16}}{2}x=2−2±−16​​
  4. 4Simplify the square root
    −16\sqrt{-16}−16​
    ↓
    4i4i4i

    The root of a negative number is imaginary: sqrt(-n) = i sqrt(n).

  5. 5The two roots
    x=−2±4i2x = \frac{-2 \pm 4i}{2}x=2−2±4i​
    ↓
    x1=−1+2 i,x2=−1−2 ix_1 = -1 + 2\,i, \quad x_2 = -1 - 2\,ix1​=−1+2i,x2​=−1−2i

Steps

  1. 1Divide by a
    x2+2x+5=0x^{2} + 2x + 5 = 0x2+2x+5=0
    ↓
    x2+2x=−5x^2 + 2x = -5x2+2x=−5

    a is already 1, so nothing changes.

  2. 2Add the square of half the x coefficient
    (b2a)2\left(\frac{b}{2a}\right)^2(2ab​)2
    ↓
    (1)2=1\left(1\right)^2 = 1(1)2=1

    Add this to both sides.

  3. 3Write the left side as a square
    x2+2x+1=−4x^2 + 2x + 1 = -4x2+2x+1=−4
    ↓
    (x+1)2=−4\left(x + 1\right)^2 = -4(x+1)2=−4

    The right side equals the discriminant over 4a squared: -16 / 4.

  4. 4Take the square root of both sides
    x+1=±−4x + 1 = \pm\sqrt{-4}x+1=±−4​
    ↓
    x+1=±2ix + 1 = \pm 2ix+1=±2i

    The right side is negative, so the roots are complex.

  5. 5Vertex form
    y=x2+2x+5y = x^{2} + 2x + 5y=x2+2x+5
    ↓
    y=(x+1)2+4y = \left(x + 1\right)^2 + 4y=(x+1)2+4

    The same completing-the-square work, kept as an expression, gives the vertex.

  1. Tools
  2. Mathematics
  3. Quadratic Formula Calculator
  4. Quadratic Equation With Complex Roots

About Quadratic Equation With Complex Roots

This calculator solves quadratics with a negative discriminant. Instead of stopping at "no real solution", it gives the two complex roots in the form a ± bi. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.

The page opens on x² + 2x + 5 = 0. The discriminant is -16, and the square root of -16 is 4i, so the roots are -1 + 2i and -1 - 2i. The graph shows why: the parabola never reaches the x axis, and its vertex is the point (-1, 4).

Complex roots of a real quadratic always come as a conjugate pair: the real part is -b/(2a) and the imaginary parts are equal with opposite signs.

How to use Quadratic Equation With Complex Roots

  1. 1Choose what to show: Roots, Discriminant, Vertex or Completing the square.
  2. 2Enter a, b and c in their boxes, or switch to Equation and type something like 2x^2 - 7x + 3 = 0.
  3. 3Read the headline answer. It updates as you type, with no calculate button.
  4. 4Open the steps underneath to follow the substitution into the quadratic formula, or the completing-the-square route.
  5. 5Use the table for the factored form, vertex form, axis of symmetry and y-intercept, and the graph to see where the parabola crosses the x axis.

Features

  • Exact roots: rational, simplified radical or complex, never only a rounded decimal.
  • Discriminant with a reading: two real roots, one repeated root or a complex pair.
  • Vertex, axis of symmetry, y-intercept and whether the parabola opens up or down.
  • Factored form over the rationals when it exists, with the complex factorisation otherwise.
  • Completing the square written out, ending in vertex form.
  • Quadratic formula steps in KaTeX, with a copy button for the LaTeX.
  • Interactive plot with roots and vertex marked, zoom, pan and PNG export.
  • Accepts integers, decimals, fractions and a full equation.

Complex roots

When the discriminant is negative, take the root of its absolute value and attach i.

x = -b/(2a) ± i·√(4ac - b²) / (2a)

Worked examples

  • x² + 2x + 5 = 0 → gives x = -1 + 2i and x = -1 - 2i

    Discriminant -16, so the imaginary part is 4/2 = 2.

  • x² + x + 1 = 0 → gives x = (-1 + √3i)/2 and x = (-1 - √3i)/2

    Discriminant -3. These roots are the primitive cube roots of 1.

Privacy: how it works

  • Everything is calculated by code running in your browser tab. The coefficients you enter are not sent to Ulkaa or to any other server, and there is nothing to sign in to.

Tips

  • Check an answer by substituting a root back into the equation: the left side should come out as 0.
  • If the roots look awkward, look at the discriminant first. A perfect square means rational roots and a clean factorisation.
  • The sum of the roots is -b/a and their product is c/a, which is a quick check on any pair of roots.

Frequently asked questions

Is it really free?

Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.

Do I need an account?

No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.

Does it run in my browser?

Yes. Coefficients are processed by code running in your browser tab, on your own device.

Is my data uploaded?

No. Coefficients are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.

Why does a quadratic have complex roots?

When the discriminant is negative the formula needs the square root of a negative number. That is defined using the imaginary unit i, where i² = -1.

Do complex roots always come in pairs?

For a quadratic with real coefficients, yes. They are complex conjugates, with the same real part and opposite imaginary parts.

More Quadratic Formula Calculator pages

  • Quadratic Formula CalculatorEnter a, b and c, or type the equation, and get the roots with every step, the vertex and a graph of the parabola.
  • Discriminant CalculatorEnter a, b and c to get the discriminant and what it says about the roots.
  • Vertex Form CalculatorEnter a, b and c to get the vertex, the axis of symmetry and the vertex form.
  • Completing the Square CalculatorEnter a, b and c to see the equation rewritten as a perfect square, step by step.

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