Quadratic Equation Calculator

To solve ax² + bx + c = 0, first compute the discriminant D = b² - 4ac. If D > 0, use the quadratic formula x = (-b ± √D)/(2a) to get two real roots. If D = 0, there is one real root x = -b/(2a). If D < 0, the roots are complex: x = (-b ± i√(-D))/(2a). For example, solve 2x² + 3x - 5 = 0: D = 9 - 4(2)(-5) = 9 + 40 = 49. Since D > 0, x = (-3 ± 7)/4, so x = 1 or x = -2.5. You can also factor when possible: 2x² + 3x - 5 = (2x + 5)(x - 1), giving the same roots. After finding roots, check them by substituting back into the original equation. For quick checks, use our percentage calculator to verify arithmetic, though it's not needed for algebra. If you need to solve many quadratics, bookmark this quadratic equation calculator for instant results.

The discriminant D = b² - 4ac tells you the number and type of roots without solving the equation. If D > 0, there are two distinct real roots, and the parabola crosses the x-axis at two points. For x² - 5x + 6 = 0, D = 25 - 24 = 1 > 0, roots are 2 and 3. If D = 0, there is exactly one real root (a double root), and the vertex lies on the x-axis. For x² - 4x + 4 = 0, D = 16 - 16 = 0, root is x = 2. If D < 0, there are two complex conjugate roots, and the parabola does not intersect the x-axis. For x² + 1 = 0, D = 0 - 4 = -4 < 0, roots are i and -i. The discriminant also indicates whether the quadratic can be factored over the integers: if D is a perfect square, it factors nicely. For example, D = 49 from 2x² + 3x - 5 = 0 is a perfect square, so it factors. Use this quadratic equation calculator to compute D and interpret results instantly.

Yes, when the discriminant D = b² - 4ac is negative, the equation has no real solutions; instead, it has two complex solutions. For example, x² + 4x + 5 = 0: D = 16 - 20 = -4 < 0. The quadratic formula gives x = (-4 ± √(-4))/2 = (-4 ± 2i)/2 = -2 ± i. So the roots are -2 + i and -2 - i. This means the graph of y = x² + 4x + 5 never crosses the x-axis; its vertex is at (-2, 1), above the axis. In real-world contexts like physics, a negative discriminant might indicate that a projectile never reaches a certain height. If you need to find where a quadratic equals a specific value other than zero, you can set up a new equation. For instance, to solve x² + 4x + 5 = 3, subtract 3 to get x² + 4x + 2 = 0, then compute D = 16 - 8 = 8 > 0, giving real roots. Always check the discriminant first. Our quadratic equation calculator handles complex roots seamlessly.

The quadratic formula is x = (-b ± √(b² - 4ac))/(2a), which solves any quadratic equation ax² + bx + c = 0. It's derived by completing the square: start with ax² + bx + c = 0, divide by a (a ≠ 0): x² + (b/a)x + c/a = 0. Move c/a to the right: x² + (b/a)x = -c/a. Add (b/(2a))² to both sides: x² + (b/a)x + b²/(4a²) = -c/a + b²/(4a²). The left side becomes (x + b/(2a))². The right side simplifies to (b² - 4ac)/(4a²). Take the square root: x + b/(2a) = ±√(b² - 4ac)/(2a). Finally, subtract b/(2a): x = (-b ± √(b² - 4ac))/(2a). For example, solve 3x² - 2x - 1 = 0: a=3, b=-2, c=-1. D = 4 + 12 = 16. x = (2 ± 4)/6, so x = 1 or x = -1/3. The formula works for all quadratics, real or complex. Use this quadratic equation calculator to avoid manual errors.

Translate the word problem into standard form ax² + bx + c = 0, then input a, b, and c into the calculator. For example, a rectangle's length is 3 more than its width, and its area is 40. Let width = w, length = w + 3. Area: w(w + 3) = 40 → w² + 3w - 40 = 0. Enter a=1, b=3, c=-40. The calculator gives D = 9 + 160 = 169, roots w = (-3 ± 13)/2 → w = 5 or w = -8. Since width can't be negative, w = 5, length = 8. Another example: a ball's height h in feet after t seconds is h = -16t² + 64t + 80. To find when it hits the ground, set h=0: -16t² + 64t + 80 = 0. Divide by -16: t² - 4t - 5 = 0. Enter a=1, b=-4, c=-5. D = 16 + 20 = 36, roots t = (4 ± 6)/2 → t = 5 or t = -1. Time can't be negative, so t = 5 seconds. Always check units and discard negative roots if they don't make sense. After solving, you might need to compute percentages, like efficiency; use our percentage calculator for that.

Common mistakes include: (1) Forgetting the negative sign in -b, e.g., for x² - 5x + 6 = 0, b = -5, so -b = 5, not -5. (2) Miscomputing the discriminant, especially with negative c: for 2x² + 3x - 5 = 0, D = 9 - 4(2)(-5) = 9 + 40 = 49, not 9 - 40. (3) Dividing by 2a incorrectly: for a=2, 2a=4, not 2. (4) Assuming all quadratics factor; many don't, like x² + x - 1 = 0, which requires the formula. (5) Ignoring complex roots when D < 0. (6) Losing a root by dividing by x: e.g., x² = 4x → x² - 4x = 0 → x(x - 4) = 0 → x = 0 or 4; dividing by x would lose x=0. (7) Not checking solutions: substitute back. For 2x² + 3x - 5 = 0, x=1 gives 2+3-5=0, correct. Use this quadratic equation calculator to verify your manual work. Also, when dealing with percentages in related problems, double-check with our percentage calculator.