Our Quadratic Formula Solver helps you quickly find the roots of any quadratic equation in the standard form ax² + bx + c = 0. Just enter the coefficients a, b, and c, and let the calculator do the work, showing both real and complex solutions.
Enter the coefficient of x² (if 0, solves as a linear equation).
Enter the coefficient of x.
Enter the constant term.
Our Quadratic Formula Solver helps you quickly find the roots of any quadratic equation in the standard form ax² + bx + c = 0. Just enter the coefficients a, b, and c, and let the calculator do the work, showing both real and complex solutions.
The quadratic formula is: x = [-b ± sqrt(b² - 4ac)] / 2a
For the equation 2x² + 5x - 3 = 0, we have a = 2, b = 5, and c = -3. 1. Calculate the discriminant: b² - 4ac = (5)² - 4(2)(-3) = 25 + 24 = 49. 2. Since the discriminant is positive (49 > 0), there are two distinct real roots. 3. Apply the quadratic formula: x = [-5 ± sqrt(49)] / (2*2) = [-5 ± 7] / 4. 4. The two roots are: x₁ = (-5 + 7) / 4 = 2 / 4 = 0.5, and x₂ = (-5 - 7) / 4 = -12 / 4 = -3.
A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. It is typically written in the standard form ax² + bx + c = 0, where 'a', 'b', and 'c' are coefficients and 'a' is not equal to zero.
The quadratic formula finds the solutions (roots) of any quadratic equation. For ax² + bx + c = 0, the roots are x = [-b ± √(b² - 4ac)] / 2a.
The discriminant is the part of the quadratic formula under the square root sign: b² - 4ac. Its value determines the nature of the roots of the quadratic equation. If positive, there are two real roots; if zero, there is one real root; if negative, there are two complex conjugate roots.
The vertex (h, k) is the turning point of the parabola y = ax² + bx + c. It is at h = −b/(2a) and k = a·h² + b·h + c. This calculator reports the vertex alongside the roots.
If a = 0 and b ≠ 0, the equation is linear (bx + c = 0) and the calculator returns x = −c/b. If both a and b are 0, it is not a valid linear or quadratic equation.
Yes, our Quadratic Formula Solver is designed to handle all types of roots, including complex conjugate roots. If the discriminant is negative, the calculator will display the solutions in the form of a + bi, where 'i' is the imaginary unit.
Quadratic Formula Solver Explained: Master Quadratic Equations
Quadratic equations are a cornerstone of algebra, appearing in everything from physics to finance. They might look intimidating with that 'x²' term, but mastering them opens up a whole new world of problem-solving. Our Quadratic Equation Solver is here to simplify the process, helping you quickly find the solutions (roots) and key features like the vertex for any equation in the standard form ax² + bx + c = 0.
A quadratic equation is a second-degree polynomial equation. This means the highest power of the variable (usually 'x') is 2. The standard form, ax² + bx + c = 0, is crucial because it sets up the equation for solving using the famous quadratic formula. Here, 'a', 'b', and 'c' are coefficients, with 'a' never being zero. If 'a' were zero, it would simply be a linear equation (bx + c = 0). The solutions to a quadratic equation are also known as its roots, and they represent the x-intercepts of the parabola when the equation is graphed.
The Quadratic Formula: x = [-b ± √(b² - 4ac)] / 2a Where: 'a' is the coefficient of x² 'b' is the coefficient of x 'c' is the constant term And the Discriminant (Δ) is: Δ = b² - 4ac Also, the Vertex of the parabola (x_v, y_v) is found using: x_v = -b / 2a y_v = a(x_v)² + b(x_v) + c
Let's work through an example: Solve the equation 2x² - 7x + 3 = 0. Here, a = 2, b = -7, and c = 3. First, calculate the discriminant: Δ = b² - 4ac = (-7)² - 4(2)(3) = 49 - 24 = 25 Since Δ > 0, we know there will be two distinct real roots. Now, apply the quadratic formula: x = [-(-7) ± √25] / (2 * 2) x = [7 ± 5] / 4 For x₁: (7 + 5) / 4 = 12 / 4 = 3 For x₂: (7 - 5) / 4 = 2 / 4 = 0.5 So, the roots are x = 3 and x = 0.5. To find the vertex: x_v = -b / 2a = -(-7) / (2 * 2) = 7 / 4 = 1.75 y_v = 2(1.75)² - 7(1.75) + 3 = 2(3.0625) - 12.25 + 3 = 6.125 - 12.25 + 3 = -3.125 The vertex is (1.75, -3.125).
Using our Quadratic Equation Solver is straightforward. Simply enter the numerical values for the coefficients 'a', 'b', and 'c' into the respective input fields. Remember that 'a' cannot be zero for it to be a quadratic equation. If a term is missing (e.g., no 'x' term), enter 0 for its coefficient. Once you've entered the values, the calculator will instantly display the roots (solutions), indicating whether they are real or complex, along with the discriminant and the coordinates of the parabola's vertex.
© 2025-2026 PromathTools. All rights reserved.
Built by KruskalCode – SaaS & Automation Experts · EverydayDevTools
