Roots of ax² + bx + c = 0.
x = (-b ± √(b² - 4ac)) / (2a)
For any quadratic equation of the form ax² + bx + c = 0 (where a ≠ 0), the two solutions for x are given by:
x = (−b ± √(b² − 4ac)) / (2a)
The expression under the square root, b² − 4ac, is called the discriminant (Δ). It tells you the nature of the roots:
Discriminant: Δ = (−4)² − 4(2)(−6) = 16 + 48 = 64 √Δ = √64 = 8 x₁ = (−(−4) + 8) / (2 × 2) = (4 + 8) / 4 = 12 / 4 = 3 x₂ = (−(−4) − 8) / (2 × 2) = (4 − 8) / 4 = −4 / 4 = −1
Result: x₁ = 3 and x₂ = −1
Because the discriminant is 64 (greater than zero), the equation has two distinct real roots: x = 3 and x = −1. You can verify: 2(3)² − 4(3) − 6 = 18 − 12 − 6 = 0 ✓ and 2(−1)² − 4(−1) − 6 = 2 + 4 − 6 = 0 ✓. On a graph, these are the two x-intercepts of the parabola y = 2x² − 4x − 6.
A quadratic equation is any polynomial equation of degree 2. In standard form it is written ax² + bx + c = 0. Quadratics appear throughout mathematics, physics (projectile motion), engineering (circuit analysis), economics (profit maximisation), and computer graphics.
The quadratic formula is derived by completing the square on the general equation ax² + bx + c = 0:
The result is the familiar formula x = (−b ± √(b² − 4ac)) / (2a).
The discriminant Δ = b² − 4ac is the most important diagnostic tool:
Without solving the full formula, you can quickly check your answers using:
These identities, known as Vieta's formulas, let you verify results instantly.
The quadratic formula always works, but for simple equations other methods may be faster:
The vertex is the minimum or maximum point of the parabola y = ax² + bx + c. Its x-coordinate is x = −b/(2a), which is exactly the midpoint of the two roots given by the quadratic formula. The y-coordinate is found by substituting this x back into the equation.
For ax² + bx + c = 0, the sum of the roots equals −b/a and the product of the roots equals c/a. After solving with the quadratic formula, add and multiply your answers and compare these values as a quick sanity check.
They refer to the same concept from different perspectives. 'Roots' or 'zeros' are the values of x that make the equation ax² + bx + c = 0 true (algebra). 'X-intercepts' are the points where the parabola y = ax² + bx + c crosses the x-axis (geometry). All three terms describe identical values.
Yes, but technically that single solution is a 'repeated root' — the same value appearing twice. It occurs when the discriminant is zero. For example, x² − 6x + 9 = 0 has the double root x = 3, since (x − 3)² = 0.
In physics, the height of a projectile is modelled as h(t) = −½gt² + v₀t + h₀. Setting h(t) = 0 and identifying a = −½g, b = v₀, c = h₀ lets you apply the quadratic formula to find the times when the object is at ground level — i.e., when it was launched and when it lands.
When b² − 4ac < 0, the square root is of a negative number, producing two complex conjugate roots of the form x = (−b ± i√|Δ|) / (2a), where i = √(−1). These roots are not real numbers and the parabola does not intersect the x-axis.
No. If a = 0, the equation reduces to bx + c = 0, which is a linear equation solved by x = −c/b. The quadratic formula requires a ≠ 0 because dividing by 2a = 0 is undefined.
Rearrange it first. For example, 3x² = 5x − 2 becomes 3x² − 5x + 2 = 0, giving a = 3, b = −5, c = 2. Always move all terms to one side before identifying the coefficients.
A double root occurs when the discriminant equals exactly zero (b² − 4ac = 0). The formula gives x = −b / (2a), a single value that is counted as two identical roots. Graphically, the parabola just touches the x-axis at its vertex without crossing it.
Yes. You can enter any real number for a, b, and c — integers, decimals, or fractions converted to decimals. The calculator will return exact or decimal roots as appropriate.
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