Quadratic Solver

How to use the Quadratic Solver

Enter three coefficients and the complete solution set appears immediately.

1. Enter coefficient aThis is the coefficient of x². It must be non-zero for the equation to be quadratic. Furthermore, if a = 0, the tool solves it as a linear equation instead.
2. Enter coefficient bThis is the coefficient of x. It can be any real number including zero.
3. Enter coefficient cThis is the constant term. Furthermore, it equals −y-intercept of the parabola when the equation represents a graph.
4. Click Solve equationThe roots appear immediately along with the discriminant and vertex. Moreover, for two real roots, Vieta's relations (sum and product) are shown for verification.
5. Interpret the discriminantPositive: two real roots. Zero: one repeated root. Negative: two complex conjugate roots. Additionally, the magnitude tells you how far apart the roots are.

Options and variants explained

The discriminant determines the nature of the roots without solving the full equation.

The formula explained

The formula derives from completing the square. Moving c to the right side, dividing by a, adding (b/2a)² to both sides and taking the square root produces the standard quadratic formula. Consequently, every quadratic has exactly two solutions — which may be equal, real or complex — as guaranteed by the fundamental theorem of algebra.

Worked example: x² − 5x + 6 = 0 (a=1, b=−5, c=6)

Discriminant: (−5)² − 4×1×6 = 25 − 24 = 1. Positive discriminant → two real roots. x₁ = (5 + √1) ÷ 2 = 3. x₂ = (5 − √1) ÷ 2 = 2. Therefore the roots are x = 3 and x = 2.

Verification using Vieta's relations: sum of roots = 3 + 2 = 5 = −b/a = −(−5)/1 ✓. Product of roots = 3 × 2 = 6 = c/a = 6/1 ✓. Furthermore, the factored form is (x − 3)(x − 2) = 0, which confirms the roots.

Complex roots example: x² + 2x + 5 = 0

Discriminant: 4 − 20 = −16. Negative → complex roots. x = (−2 ± √(−16)) ÷ 2 = −1 ± 2i. The roots are −1 + 2i and −1 − 2i. Moreover, complex roots always appear as conjugate pairs when a, b and c are real numbers.

What is a quadratic equation?

A quadratic equation is any polynomial equation of degree 2 — the highest power of the variable is 2. The standard form is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. Furthermore, the word quadratic comes from the Latin quadratus, meaning square.

Quadratic equations model many physical phenomena. Projectile motion — the path of a ball through the air — follows a parabolic trajectory described by a quadratic. Additionally, economics uses quadratic cost functions, optics uses parabolic reflectors and engineering uses quadratic constraints in optimisation problems.

Every quadratic has exactly two roots when counted with multiplicity. This follows from the fundamental theorem of algebra. Moreover, when the roots are complex, they always form a conjugate pair — one is the complex conjugate of the other — because the coefficients are real.

Why the discriminant matters

The discriminant b² − 4ac is the key quantity in a quadratic. It determines whether the equation has solutions in the real numbers before any calculation is done. Furthermore, checking the sign of the discriminant is always the first step in solving a quadratic.

In geometry, the discriminant tells you how the parabola y = ax² + bx + c relates to the x-axis. Positive discriminant: parabola crosses the x-axis at two points. Zero: tangent at one point. Negative: parabola is entirely above or below the x-axis. Moreover, the axis of symmetry is always at x = −b/(2a) regardless of the discriminant value.

Common quadratic solving mistakes

Forgetting to include the ± when taking the square root of the discriminant produces only one root instead of two. Both the + and − cases must be evaluated separately. Furthermore, this is why the quadratic formula has ± explicitly written rather than just +.

Dividing by 2a before computing the discriminant introduces errors if the order of operations is incorrect. The numerator −b ± √(b²−4ac) must be computed entirely before dividing by 2a. Moreover, entering this incorrectly into a calculator is a common exam error.

Assuming complex roots mean no solution is incorrect. Complex roots are valid solutions — they just lie in the complex plane rather than on the real number line. Consequently, every quadratic has exactly two roots; none are "missing" when the discriminant is negative.

Tips for solving quadratics efficiently

Try factoring first for integer coefficients. If the product of a and c has factor pairs that sum to b, the equation factors cleanly without the formula. For example, x² − 5x + 6 = 0 factors to (x−3)(x−2) = 0 by inspection. Furthermore, factoring is faster than the formula when it works.

Use Vieta's formulas to verify roots without substituting back into the equation. The sum of roots must equal −b/a and the product must equal c/a. Moreover, these checks are faster than plugging each root back into the original equation.

For approximate roots, the discriminant's magnitude gives the root separation. Roots are approximately −b/(2a) ± √Δ/(2|a|). Additionally, when the discriminant is small relative to b², the two roots lie close together near the vertex x-coordinate.

Frequently asked questions

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