Pre Calc Calculator

Quadratic Equation Solver

Enter the coefficients a, b, and c for a quadratic equation in the form ax² + bx + c = 0 to find its roots.

Understanding the Quadratic Equation and Its Roots

In pre-calculus, one of the fundamental concepts you'll encounter is the quadratic equation. 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.
• a, b, and c are coefficients, with a not equal to zero. If a were zero, the equation would become a linear equation (bx + c = 0).

What are Roots?

The "roots" (also known as "solutions" or "zeros") of a quadratic equation are the values of x that satisfy the equation, meaning when you substitute these values into the equation, the equation holds true (it equals zero). Graphically, these roots represent the x-intercepts of the parabola that the quadratic equation forms when plotted on a coordinate plane.

The Quadratic Formula

While some quadratic equations can be solved by factoring or completing the square, the quadratic formula is a universal method that works for all quadratic equations. It provides a direct way to find the roots:

x = [-b ± √(b² - 4ac)] / 2a

This formula allows us to calculate two potential roots, often denoted as x₁ and x₂, due to the "±" (plus or minus) sign.

The Discriminant (b² – 4ac)

A crucial part of the quadratic formula is the expression under the square root, b² - 4ac. This is called the "discriminant" (often denoted by the Greek letter Delta, Δ). The value of the discriminant tells us about the nature of the roots:

• If Δ > 0: There are two distinct real roots. This means the parabola intersects the x-axis at two different points.
• If Δ = 0: There is exactly one real root (a repeated root). This means the parabola touches the x-axis at exactly one point (its vertex).
• If Δ < 0: There are two complex conjugate roots. This means the parabola does not intersect the x-axis at all. The roots involve the imaginary unit i, where i = √(-1).

How to Use the Quadratic Equation Solver

Our calculator simplifies the process of finding these roots. Simply input the numerical values for the coefficients a, b, and c from your quadratic equation ax² + bx + c = 0 into the respective fields. Click "Calculate Roots," and the calculator will instantly provide the solutions, indicating whether they are real or complex.

Example Scenarios:

Example 1: Two Distinct Real Roots

Consider the equation: x² - 3x + 2 = 0

• a = 1
• b = -3
• c = 2

Using the calculator with these values will yield: x₁ = 2 and x₂ = 1.

Example 2: One Real (Repeated) Root

Consider the equation: x² - 4x + 4 = 0

• a = 1
• b = -4
• c = 4

Using the calculator with these values will yield: x = 2.

Example 3: Two Complex Conjugate Roots

Consider the equation: x² + 2x + 5 = 0

• a = 1
• b = 2
• c = 5

Using the calculator with these values will yield: x₁ = -1 + 2i and x₂ = -1 - 2i.

This calculator is a valuable tool for students and professionals alike, helping to quickly verify solutions and deepen understanding of quadratic equations in pre-calculus and beyond.