Math Problem Calculator

Quadratic Equation Solver

x = " + x.toFixed(4) + "

x₁ = " + x1.toFixed(4) + "

x₂ = " + x2.toFixed(4) + "

x = " + x.toFixed(4) + "

x₁ = " + realPart.toFixed(4) + " + " + imaginaryPart.toFixed(4) + "i

x₂ = " + realPart.toFixed(4) + " – " + imaginaryPart.toFixed(4) + "i

Understanding the Quadratic Equation and Its Solutions

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:

ax² + bx + c = 0

Where:

• a, b, and c are coefficients (constants), with 'a' not equal to zero.
• x represents the unknown variable.

The solutions to a quadratic equation are also known as its roots. These roots are the values of 'x' that satisfy the equation, making the entire expression equal to zero. A quadratic equation can have two distinct real roots, one repeated real root, or two complex conjugate roots.

The Quadratic Formula

The most common method to find the roots of a quadratic equation is by using the quadratic formula:

x = [-b ± √(b² – 4ac)] / 2a

This formula directly provides the values of 'x' once the coefficients 'a', 'b', and 'c' are known.

The Discriminant (b² – 4ac)

A crucial part of the quadratic formula is the expression under the square root, known as the discriminant (often denoted by the Greek letter Delta, Δ):

Δ = b² – 4ac

The value of the discriminant determines the nature of the roots:

• If Δ > 0 (Discriminant is positive): The equation has two distinct real roots. This means there are two different numerical values for 'x' that satisfy the equation. Graphically, the parabola intersects the x-axis at two different points.
• If Δ = 0 (Discriminant is zero): The equation has exactly one real root (also called a repeated or double root). This means there is only one numerical value for 'x' that satisfies the equation. Graphically, the parabola touches the x-axis at exactly one point (its vertex).
• If Δ < 0 (Discriminant is negative): The equation has two complex conjugate roots. These roots involve the imaginary unit 'i' (where i² = -1) and do not appear on the real number line. Graphically, the parabola does not intersect the x-axis.

How to Use the Quadratic Equation Solver

Our Quadratic Equation Solver simplifies the process of finding the roots:

1. Enter Coefficient 'a': Input the number that multiplies x². Remember, 'a' cannot be zero for it to be a quadratic equation. If 'a' is 0, the calculator will treat it as a linear equation (bx + c = 0).
2. Enter Coefficient 'b': Input the number that multiplies x.
3. Enter Coefficient 'c': Input the constant term.
4. Click "Calculate Roots": The calculator will instantly apply the quadratic formula and display the roots, indicating whether they are real or complex.

Examples

Let's look at some practical examples:

Example 1: Two Distinct Real Roots

Consider the equation: x² – 5x + 6 = 0

• a = 1
• b = -5
• c = 6

Using the calculator with these values will yield:

x₁ = 3.0000

x₂ = 2.0000

Here, Δ = (-5)² – 4(1)(6) = 25 – 24 = 1 (positive), indicating two distinct real roots.

Example 2: One Real Root (Repeated)

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

• a = 1
• b = 4
• c = 4

Using the calculator with these values will yield:

x = -2.0000

Here, Δ = (4)² – 4(1)(4) = 16 – 16 = 0 (zero), indicating one repeated real root.

Example 3: Two Complex Conjugate Roots

Consider the equation: x² + x + 1 = 0

• a = 1
• b = 1
• c = 1

Using the calculator with these values will yield:

x₁ = -0.5000 + 0.8660i

x₂ = -0.5000 – 0.8660i

Here, Δ = (1)² – 4(1)(1) = 1 – 4 = -3 (negative), indicating two complex conjugate roots.

This calculator is a handy tool for students, engineers, and anyone needing to quickly solve quadratic equations without manual calculation, providing accurate results for all types of roots.