Quadratic Equation Solver
Enter the coefficients for your quadratic equation in the form ax² + bx + c = 0 to find its roots.
Understanding the Quadratic Equation and Its Roots
A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term in which the unknown variable is raised to the power of two. The standard form of a quadratic equation is expressed as ax² + bx + c = 0, where 'x' represents the unknown variable, and 'a', 'b', and 'c' are coefficients, with 'a' not equal to zero.
The Quadratic Formula
The most common method for finding the roots (or solutions) of a quadratic equation is by using the quadratic formula. This formula provides a direct way to calculate the values of 'x' that satisfy the equation. The formula is:
x = [-b ± sqrt(b² - 4ac)] / 2a
Here, the '±' symbol indicates that there will generally be two solutions for 'x', one using the plus sign and one using the minus sign.
The Discriminant (b² – 4ac)
A crucial part of the quadratic formula is the expression under the square root, b² - 4ac. This is known as the discriminant, and its value determines the nature of the roots:
- If the discriminant is positive (
b² - 4ac > 0): The equation has two distinct real roots. This means there are two different numerical values for 'x' that will satisfy the equation. Graphically, the parabola representing the quadratic equation will intersect the x-axis at two different points. - If the discriminant is zero (
b² - 4ac = 0): The equation has exactly one real root (sometimes called a repeated or double root). In this case, the two solutions from the quadratic formula become identical. Graphically, the parabola touches the x-axis at exactly one point. - If the discriminant is negative (
b² - 4ac < 0): The equation has two complex (or imaginary) roots. These roots will be conjugates of each other, meaning they will have a real part and an imaginary part. Graphically, the parabola does not intersect the x-axis at all.
How Texas Instruments Calculators Solve Quadratic Equations
Texas Instruments (TI) calculators, especially graphing calculators like the TI-83, TI-84, or TI-Nspire, are powerful tools for solving quadratic equations. While you can manually input the coefficients into the quadratic formula, these calculators often have built-in functions or apps to streamline the process:
- Polynomial Root Finder: Many TI calculators include a "Polynomial Root Finder" or "Poly Smlt" (Polynomial Simultaneous Equation Solver) application. You would typically select the degree of the polynomial (2 for quadratic), input the coefficients 'a', 'b', and 'c', and the calculator would instantly display the roots, including complex solutions if applicable.
- Graphing: You can also graph the quadratic function
y = ax² + bx + cand use the calculator's "zero" or "root" function to find where the graph intersects the x-axis. This visually confirms the real roots. - Equation Solver: Some advanced TI calculators have a general equation solver where you can type in the equation
ax² + bx + c = 0and let the calculator solve for 'x'.
Our online Quadratic Equation Solver mimics the core functionality of these TI calculator features, allowing you to quickly find the roots of any quadratic equation by simply entering its coefficients.
Examples:
- Example 1: Two Real Roots
Equation:x² - 5x + 6 = 0
Coefficients: a=1, b=-5, c=6
Discriminant: (-5)² – 4(1)(6) = 25 – 24 = 1 (positive)
Roots: x₁ = [5 + sqrt(1)] / 2 = 3, x₂ = [5 – sqrt(1)] / 2 = 2 - Example 2: One Real Root (Repeated)
Equation:x² - 4x + 4 = 0
Coefficients: a=1, b=-4, c=4
Discriminant: (-4)² – 4(1)(4) = 16 – 16 = 0 (zero)
Root: x = [4 + sqrt(0)] / 2 = 2 - Example 3: Two Complex Roots
Equation:x² + 2x + 5 = 0
Coefficients: a=1, b=2, c=5
Discriminant: (2)² – 4(1)(5) = 4 – 20 = -16 (negative)
Roots: x₁ = [-2 + sqrt(-16)] / 2 = [-2 + 4i] / 2 = -1 + 2i
x₂ = [-2 – sqrt(-16)] / 2 = [-2 – 4i] / 2 = -1 – 2i