Quadratic Equation Solver
Enter the coefficients a, b, and c for the quadratic equation in the standard form ax² + bx + c = 0 to find its solutions (roots).
Understanding Quadratic Equations
A quadratic equation is a polynomial equation of the second degree. The general form is written as:
ax² + bx + c = 0
where x represents an unknown variable, and a, b, and c are numerical coefficients. A critical condition for an equation to be quadratic is that the coefficient a must not be equal to zero. If a were zero, the ax² term would vanish, and the equation would simplify to a linear equation (bx + c = 0).
The Quadratic Formula
The solutions (also known as roots) for x in a quadratic equation can be found using the quadratic formula. This formula is derived by completing the square and provides a direct method to calculate the values of x that satisfy the equation:
x = [-b ± √(b² - 4ac)] / 2a
The '±' symbol indicates that there are generally two solutions, one using the plus sign and one using the minus sign.
The Discriminant (b² – 4ac)
The term inside the square root, b² - 4ac, is of particular importance and is called the discriminant (often denoted by the Greek letter Delta, Δ). Its value determines the nature of the solutions:
- If Δ > 0 (Positive Discriminant): There are two distinct real solutions. This means the graph of the quadratic equation (a parabola) intersects the x-axis at two different points.
- If Δ = 0 (Zero Discriminant): There is exactly one real solution. This is sometimes referred to as a repeated root or two identical real solutions. Graphically, the parabola touches the x-axis at exactly one point (its vertex lies on the x-axis).
- If Δ < 0 (Negative Discriminant): There are two complex conjugate solutions. In this case, the parabola does not intersect the x-axis; its vertex is either entirely above or entirely below the x-axis. Complex solutions involve the imaginary unit 'i', where
i = √(-1).
How to Use This Calculator
- Identify Coefficients: Begin by ensuring your quadratic equation is in the standard form
ax² + bx + c = 0. Then, identify the numerical values fora,b, andc. Pay close attention to their signs (positive or negative). - Enter Values: Input these numerical values into the corresponding fields: 'Coefficient a', 'Coefficient b', and 'Coefficient c'.
- Calculate: Click the "Calculate Solutions" button.
- View Results: The calculator will instantly display the solutions for
x, clearly indicating whether they are real or complex, and if there are one or two distinct solutions.
Examples
Let's explore a few examples to illustrate the different types of solutions:
Example 1: Two Distinct Real Solutions
Consider the equation: x² - 3x + 2 = 0
a = 1b = -3c = 2
Using the calculator, you would input 1, -3, and 2. The discriminant would be (-3)² – 4(1)(2) = 9 – 8 = 1 (which is > 0).
Solutions: x₁ = 2, x₂ = 1
Example 2: One Real Solution (Repeated Root)
Consider the equation: x² + 4x + 4 = 0
a = 1b = 4c = 4
Inputting these values (1, 4, 4) into the calculator. The discriminant would be (4)² – 4(1)(4) = 16 – 16 = 0.
Solution: x = -2
Example 3: Two Complex Conjugate Solutions
Consider the equation: x² + x + 1 = 0
a = 1b = 1c = 1
Entering 1, 1, and 1 into the calculator. The discriminant would be (1)² – 4(1)(1) = 1 – 4 = -3 (which is < 0).
Solutions: x₁ = -0.5 + 0.866025i, x₂ = -0.5 – 0.866025i
Example 4: Linear Equation (when 'a' is 0)
Consider the equation: 0x² + 5x - 10 = 0 (which simplifies to 5x - 10 = 0)
a = 0b = 5c = -10
Inputting 0, 5, and -10. The calculator will recognize that 'a' is 0 and solve it as a linear equation.
Solution: x = 2