Quadratic Formula Solver

To use the quadratic formula solver, enter the three coefficients from your quadratic equation in the form ax² + bx + c = 0:

1. Coefficient a: Enter the coefficient of x². This cannot be zero (if it is, the equation is linear, not quadratic). For example, in 2x² + 3x - 5 = 0, a = 2.
2. Coefficient b: Enter the coefficient of x (the middle term). This can be positive, negative, or zero. In 2x² + 3x - 5 = 0, b = 3. If your equation does not have an x term, enter 0.
3. Constant c: Enter the constant term (the number without any variable). In 2x² + 3x - 5 = 0, c = -5. Remember to include the sign.

Once you have entered the coefficients, the calculator instantly provides six results:

• First Root (x₁): One solution to the equation. If there are two real roots, this is typically the larger value.
• Second Root (x₂): The other solution. For a repeated root, this equals x₁.
• Discriminant: The value b² - 4ac. A positive discriminant means two real roots, zero means one repeated root, and negative means two complex roots.
• Vertex X-Coordinate: The x-value of the parabola peak or valley, calculated as -b / 2a.
• Vertex Y-Coordinate: The y-value at the vertex, showing the extreme value of the function.
• Root Type: A description of whether the roots are real and distinct, real and repeated, or complex.

Use these results to understand the behavior of your quadratic function and to answer whatever question prompted you to solve the equation.

The quadratic formula solver is a calculator that solves any quadratic equation of the form ax² + bx + c = 0 by computing the roots using the quadratic formula. A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2. Quadratic equations appear throughout mathematics, physics, engineering, and many real-world applications, from calculating projectile motion to optimizing profit in business. The quadratic formula provides a reliable method that works for all quadratic equations, unlike factoring or completing the square, which do not always work or are sometimes difficult to apply.

The roots (also called solutions or zeros) of a quadratic equation are the x-values that make the equation true. Graphically, they are the points where the parabola (the graph of a quadratic function) crosses the x-axis. Every quadratic equation has either two real roots, one repeated real root, or two complex conjugate roots, depending on the discriminant. The discriminant (b² - 4ac) is the expression under the square root in the quadratic formula and determines which type of roots you have without needing to fully solve the equation.

The parabola itself is an important feature of quadratic equations. The vertex is the peak or valley of the parabola. A parabola with a positive coefficient a opens upward with a minimum vertex, while a parabola with negative a opens downward with a maximum vertex. Understanding the vertex and the roots together gives a complete picture of the quadratic function: where it crosses the x-axis and where its extreme value occurs. The axis of symmetry passes through the vertex and divides the parabola into two mirror-image halves.

Real-world applications of quadratic equations are extensive. In physics, the position of a falling object is described by a quadratic equation in time. In engineering, quadratic equations model stress and strain in materials. In business, profit functions are often quadratic, with roots representing break-even points. In agriculture, yield-versus-fertilizer relationships often follow quadratic patterns. Learning to solve quadratic equations unlocks the ability to model and analyze these real phenomena mathematically.

The quadratic formula solver uses the following formulas and calculations:

• The Quadratic Formula: x = (-b ± √(b² - 4ac)) / 2a. This formula solves any equation of the form ax² + bx + c = 0. The ± symbol indicates two calculations: one using addition (giving x₁) and one using subtraction (giving x₂).
• The Discriminant: Δ = b² - 4ac. This value determines root type: if Δ > 0, two real roots; if Δ = 0, one repeated root; if Δ < 0, two complex roots.
• Vertex X-Coordinate: x_v = -b / 2a. The vertex is always located at this x-value, which is exactly halfway between the two roots (when two real roots exist).
• Vertex Y-Coordinate: Substitute x_v back into the original quadratic: y_v = a(x_v)² + b(x_v) + c. This gives the function value at the extreme point.
• Complex Root Format: When the discriminant is negative, roots are expressed as a ± bi, where a is the real part and bi is the imaginary part. The two roots are always complex conjugates.

Example 1 - Two Real Roots: Solve x² - 5x + 6 = 0. Here, a = 1, b = -5, c = 6. Discriminant: (-5)² - 4(1)(6) = 25 - 24 = 1 (positive, so two real roots). Using the formula: x = (5 ± √1) / 2 = (5 ± 1) / 2. So x₁ = (5 + 1) / 2 = 3 and x₂ = (5 - 1) / 2 = 2. Vertex: x_v = 5 / 2 = 2.5, y_v = 1(2.5)² - 5(2.5) + 6 = 6.25 - 12.5 + 6 = -0.25. The parabola touches its minimum at (2.5, -0.25) and crosses the x-axis at x = 2 and x = 3.

Example 2 - One Repeated Root: Solve x² - 4x + 4 = 0. Here, a = 1, b = -4, c = 4. Discriminant: (-4)² - 4(1)(4) = 16 - 16 = 0 (zero, so one repeated root). Using the formula: x = (4 ± √0) / 2 = 4 / 2 = 2. Both x₁ and x₂ equal 2. Vertex: x_v = 4 / 2 = 2, y_v = 1(2)² - 4(2) + 4 = 4 - 8 + 4 = 0. The vertex is exactly on the x-axis at (2, 0). This parabola just touches the x-axis at one point.

Example 3 - Two Complex Roots: Solve x² + 2x + 5 = 0. Here, a = 1, b = 2, c = 5. Discriminant: (2)² - 4(1)(5) = 4 - 20 = -16 (negative, so complex roots). Using the formula: x = (-2 ± √(-16)) / 2 = (-2 ± 4i) / 2 = -1 ± 2i. So x₁ = -1 + 2i and x₂ = -1 - 2i. Vertex: x_v = -2 / 2 = -1, y_v = 1(-1)² + 2(-1) + 5 = 1 - 2 + 5 = 4. The vertex is at (-1, 4). Since a > 0 and the vertex has y > 0, the entire parabola is above the x-axis and never crosses it, which explains the complex roots.