Quadratic Equation Solver

Why use this calculator

How to use it

What each result means

x₁

The larger of the two solutions, or the root with the positive imaginary part when the roots are complex.

x₂

The smaller of the two solutions, or the conjugate root with the negative imaginary part.

x

The single solution, either the repeated root of a perfect square or the root of a linear equation.

Factored form

The equation written as a times two brackets, one for each root. Expanding the brackets gives the original equation back, which makes it a quick check on your working.

Solution

A message shown when the coefficients describe no equation at all, such as every value working or none working.

Discriminant

The value of b² − 4ac, which decides whether the roots are two real, one repeated, or a complex pair.

Nature

A plain-language reading of the discriminant, saying how many roots exist and whether they are real.

Vertex (turning point)

The coordinates of the minimum or maximum of the parabola, which is the answer to most optimisation problems.

Axis of symmetry

The vertical line through the vertex; the two roots always sit at equal distances either side of it.

Y-intercept

Where the curve crosses the vertical axis, which is always the constant term c, at x equal to zero.

Equation

Your coefficients written back out in standard form, so you can confirm the signs were entered as intended.

The formula

How this number is used in practice

Worked examples

• x² − 5x + 6 = 0, a textbook factorising case

  Discriminant 1, roots x = 3 and x = 2, vertex at (2.5, -0.25).

• x² − 4x + 4 = 0, a perfect square

  Discriminant 0 and a single repeated root at x = 2, where the curve touches the axis.

• x² + 1 = 0, with no real solution

  Discriminant -4.000000 and complex roots 0.000000 ± 1.000000i, so the curve never meets the x-axis.

• A ball thrown upward: -4.9t² + 20t + 1.5 = 0

  It lands at t = 4.155303 seconds; the other root, -0.073670, is negative and is discarded.

When people use it

• Checking algebra homework where the answer must be exact rather than approximate
• Finding when a projectile reaches the ground and how high it goes at its peak
• Locating the price or output level that maximises revenue or minimises cost
• Solving geometry problems where a fixed area constrains an unknown dimension
• Identifying whether a design equation has any real solution before building around it

Tips

• Rearrange to ax² + bx + c = 0 before reading off the coefficients; the formula is meaningless on an equation that is not set to zero.
• Carry the sign with each coefficient. Entering b as 5 instead of −5 flips both roots and is the most common single error.
• Check the discriminant before anything else. It tells you how many real roots exist without solving.
• Verify a root by substituting it back in; a correct root makes the left-hand side exactly zero.
• If a is negative the parabola opens downward, so the vertex is a maximum and the larger root is still reported first.
• Complex roots always arrive in conjugate pairs with the same real part, so seeing only one complex root means an arithmetic slip.

What it assumes, and where it stops being accurate

• Coefficients are real numbers; equations with complex coefficients are outside the scope of the standard formula used here.
• When a is zero the equation is not quadratic, and the calculator solves the linear case instead rather than returning an error symbol.
• Roots are computed in double-precision floating point, so a value that should be exactly zero may display as a very small number.
• A numerically stable rearrangement of the formula is used, but extremely large coefficient ratios can still lose precision.
• Every root is reported, including negative ones that a physical problem would reject as meaningless lengths or times.
• The plotting table shows nine points around the vertex and is a sketching aid, not a substitute for a full graph.

Terms used on this page

Discriminant

The expression b² − 4ac, whose sign determines whether the roots are real, repeated or complex.

Root

A value of x that makes the expression equal zero, shown on a graph as a crossing of the x-axis.

Vertex

The turning point of a parabola, at x = −b ÷ 2a, where the expression takes its maximum or minimum.

Axis of symmetry

The vertical line through the vertex about which the parabola is a mirror image of itself.

Complex conjugate

A pair of numbers sharing a real part with opposite imaginary parts, which is how complex roots always occur.

Questions people ask

What does the discriminant actually tell me?

It is the quantity b² − 4ac under the square root, and its sign decides the whole outcome. Positive means the parabola crosses the x-axis twice, so there are two distinct real roots. Zero means it just touches the axis at the vertex, giving one repeated root. Negative means it never reaches the axis, so both roots are complex.

Why do I get complex roots and what do they mean?

Complex roots appear when the discriminant is negative, meaning the parabola never crosses the horizontal axis. In a physical problem that usually means the situation described cannot happen, such as a projectile that never reaches a given height. In electrical engineering and control theory complex roots are meaningful and describe oscillating behaviour.

Can I always factorise instead of using the formula?

Only when the roots happen to be rational, which is true in textbooks and rarely true in practice. Factorising x² − 5x + 6 into (x − 3)(x − 2) is quick, but as soon as the discriminant is not a perfect square the factors are irrational and the formula is the only reliable route. The formula always works, including when factorising would.

What is the vertex used for?

It is the turning point of the parabola and therefore the maximum or minimum value the expression can take. When a quadratic models revenue, height or cost, the vertex answers the real question, which is usually how high, how much or when the peak occurs, rather than where the quantity hits zero.

What happens if a is zero?

The x² term disappears and the equation becomes linear, bx + c = 0, with the single solution x = −c ÷ b. There is no parabola, no vertex and no discriminant. The calculator detects this and solves the linear case rather than dividing by zero, and it labels the result so you know the equation was not a quadratic.

Why do the two roots sometimes look asymmetric?

They are always symmetric about the axis of symmetry at x = −b ÷ 2a, but that line is rarely at zero, so the roots look unequal in magnitude. Their average is always exactly the vertex x-coordinate, and their sum is always −b ÷ a while their product is c ÷ a. Those two relationships are a quick check on any answer.

Use this with

Quadratic Equation Solver answers one part of the question. These pick up where it stops: Fraction Calculator, Pythagorean Theorem Calculator, Percentage Calculator and Mean, Median, Mode Calculator.