Solve any quadratic equation of the form ax² + bx + c = 0. Find both roots, discriminant, and vertex with step-by-step solution.
A Quadratic Calculator solves quadratic equations of the form ax² + bx + c = 0, finding the value or values of x that satisfy the equation. This tool is fundamental in algebra, used for solving problems involving projectile motion, area optimization, and any relationship where a quantity depends on the square of a variable.
The quadratic formula states: x = [−b ± √(b² − 4ac)] / 2a, where a, b, and c are the coefficients from the standard quadratic equation form. The term under the square root (b² − 4ac) is called the discriminant, and its sign determines the nature of the solutions: positive gives two real solutions, zero gives exactly one repeated real solution, and negative gives two complex (non-real) solutions.
Enter the coefficients a, b, and c from your quadratic equation. The calculator applies the quadratic formula and returns the solution(s) for x, clearly indicating whether the equation has two real solutions, one repeated solution, or complex solutions based on the discriminant.
Example 1: For the equation x² − 5x + 6 = 0 (a=1, b=−5, c=6), the discriminant is (−5)² − 4(1)(6) = 25−24 = 1, giving two real solutions: x = (5±1)/2, which simplifies to x = 3 and x = 2.
Example 2: For the equation x² + 4x + 4 = 0 (a=1, b=4, c=4), the discriminant is 4² − 4(1)(4) = 16−16 = 0, giving exactly one repeated solution: x = −4/2 = −2.
The discriminant (b² − 4ac) determines the nature of the solutions without needing to fully solve the equation: a positive value means two distinct real solutions, zero means one repeated real solution (the parabola touches the x-axis at exactly one point), and a negative value means two complex solutions (the parabola never crosses the x-axis).
The solutions to a quadratic equation ax² + bx + c = 0 correspond exactly to the x-intercepts (where the graph crosses the x-axis) of the parabola formed by y = ax² + bx + c, making the quadratic formula a way to find these specific graphical points algebraically.
Factoring rewrites a quadratic expression as a product of two simpler binomial expressions (like (x−2)(x−3) instead of x²−5x+6), and setting each factor equal to zero provides an alternative solving method that works well when the equation factors neatly with whole numbers.
Many quadratic equations don't factor neatly into whole numbers, making the quadratic formula a universally applicable method that works for any quadratic equation, regardless of whether the coefficients produce clean, easily factorable numbers.
Projectile motion, such as calculating when a thrown object will hit the ground, is commonly modeled using a quadratic equation, since the object's height changes according to a quadratic relationship with time due to the constant acceleration of gravity.
A negative discriminant means the parabola never touches the x-axis, so there are no real number solutions; the solutions instead involve the imaginary unit i (the square root of −1), representing complex numbers that are meaningful in advanced mathematics and engineering but don't correspond to a real-world x-intercept.
The quadratic formula is actually derived by applying the completing the square technique to the general quadratic equation, meaning both methods are mathematically connected, with the formula essentially providing a pre-solved shortcut for the completing the square process.
Yes, when the discriminant equals exactly zero, the equation has precisely one repeated real solution, corresponding graphically to the parabola's vertex touching the x-axis at a single point rather than crossing through two distinct points.
Problems asking for the maximum or minimum value of a quadratic expression, such as the largest possible rectangular area given a fixed perimeter, use the vertex of the parabola (related to but distinct from the roots found by the quadratic formula) to identify the optimal solution.
Different methods offer different advantages depending on the specific equation, factoring is often fastest for simple cases, while the quadratic formula reliably works for every case, so learning multiple approaches builds flexible problem-solving skills across different types of quadratic equations.
The vertex x-coordinate is found using −b/2a, a formula derived from the symmetry of the parabola around its turning point, which is distinct from but related to the root-finding quadratic formula itself.