Quadratic Formula Calculator
Solve quadratic equations using the quadratic formula with step-by-step solutions
Quadratic Equation
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Solution
Understanding the Quadratic Formula
A quadratic equation is a second-order polynomial equation in a single variable, standardly written in the form:ax² + bx + c = 0Where x represents the unknown variable, and a, b, and c are constant coefficients (with a ≠ 0). The quadratic formula is a universal algebraic tool used to find the roots (solutions) of any quadratic equation.
The Quadratic Formula
The solutions of a quadratic equation are given by the formula:x = (-b ± √(b² - 4ac)) / 2aThe symbol ± indicates that there are generally two solutions: one found by adding the square root term, and another by subtracting it.
The Discriminant (Δ) and Root Types
The term inside the square root, b² - 4ac, is called the discriminant (denoted by Δ). It determines the number and nature of the equation's roots:
- Δ > 0 (Positive Discriminant): The equation has two distinct real roots. The graph of the parabola crosses the x-axis at two separate points.
- Δ = 0 (Zero Discriminant): The equation has one repeated real root (also called a double root). The vertex of the parabola touches the x-axis at exactly one point.
- Δ < 0 (Negative Discriminant): The equation has two complex conjugate roots (containing the imaginary unit i). The graph of the parabola does not cross or touch the x-axis.
How to Solve a Quadratic Equation Step-by-Step
Let's walk through solving the quadratic equation: x² - 5x + 6 = 0.
- Identify the coefficients: Here,
a = 1,b = -5, andc = 6. - Calculate the discriminant (Δ):
Δ = b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1. SinceΔ > 0, we have two real roots. - Apply the quadratic formula:
x = (-(-5) ± √1) / (2 × 1)x = (5 ± 1) / 2 - Find the two solutions:
x₁ = (5 + 1) / 2 = 6 / 2 = 3x₂ = (5 - 1) / 2 = 4 / 2 = 2
The roots of the equation are x = 3 and x = 2.
Frequently Asked Questions (FAQ)
What is the vertex of a parabola?
The vertex represents the highest or lowest point (turning point) of the parabola. Its x-coordinate is given by x = -b / 2a, and the y-coordinate is found by substituting this x-value back into the quadratic equation.
Can "a" be zero in a quadratic equation?
No. If a = 0, the ax² term disappears, and the equation becomes a first-order linear equation (bx + c = 0) rather than a quadratic equation.
How do you solve quadratic equations by factoring?
Factoring involves rewriting the equation as a product of two linear terms, e.g., (x - r₁)(x - r₂) = 0. According to the zero-product property, the equation is solved when either term is zero, meaning x = r₁ or x = r₂. Factoring only works easily when the roots are rational numbers.