Ti Calculator Online

Quadratic Equation Solver

Understanding Quadratic Equations and Their Solutions with an Online TI Calculator

Quadratic equations are fundamental in algebra and appear in various fields, from physics to engineering and economics. A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term in which the unknown variable is squared, but no term with a higher power. The standard form of a quadratic equation is:

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 simplify to bx + c = 0, which is a linear equation, not a quadratic one.

The Quadratic Formula: Your Key to Solutions

Solving quadratic equations manually can sometimes be complex, especially when factoring isn't straightforward. This is where the quadratic formula becomes invaluable. It provides a direct method to find the roots (or solutions) of any quadratic equation. The formula is:

x = [-b ± √(b² - 4ac)] / 2a

The term inside the square root, (b² - 4ac), is called the discriminant, often denoted by Δ or D. The value of the discriminant determines the nature of the roots:

• If D > 0 (positive discriminant): There are two distinct real roots. This means the parabola (the graph of a quadratic equation) intersects the x-axis at two different points.
• If D = 0 (zero discriminant): There is exactly one real root (also called a repeated root or a double root). The parabola touches the x-axis at exactly one point.
• If D < 0 (negative discriminant): There are two complex conjugate roots. The parabola does not intersect the x-axis. These roots involve the imaginary unit i, where i = √(-1).

How to Use This Online TI Calculator for Quadratic Equations

Our online TI-style calculator simplifies the process of finding the roots of quadratic equations. Instead of manually plugging values into the quadratic formula, you can quickly get the solutions. Here's how:

1. Identify Coefficients: Look at your quadratic equation and identify the values for a, b, and c. Remember, the equation must be in the form ax² + bx + c = 0.
2. Enter Values: Input the numerical values for 'Coefficient a', 'Coefficient b', and 'Coefficient c' into the respective fields in the calculator above.
3. Calculate: Click the "Calculate Roots" button.
4. View Results: The calculator will instantly display the roots of your equation, indicating whether they are real or complex.

Examples of Quadratic Equation Solutions

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, the discriminant D = (-5)² - 4(1)(6) = 25 - 24 = 1, which is positive, indicating two real roots.

Example 2: One Real Root (Repeated)

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

• a = 1
• b = -4
• c = 4

Inputting these values into the calculator will show:

x = 2.0000

In this case, the discriminant D = (-4)² - 4(1)(4) = 16 - 16 = 0, resulting in one repeated real root.

Example 3: Two Complex Conjugate Roots

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

• a = 1
• b = 2
• c = 5

The calculator will provide:

x₁ = -1.0000 + 2.0000i
x₂ = -1.0000 - 2.0000i

Here, the discriminant D = (2)² - 4(1)(5) = 4 - 20 = -16, which is negative, leading to two complex conjugate roots.

Example 4: Linear Equation (a=0)

Consider the equation: 0x² + 3x - 9 = 0 (or simply 3x - 9 = 0)

• a = 0
• b = 3
• c = -9

The calculator will correctly identify this as a linear equation and provide:

This is a linear equation (a=0). The solution is x = 3.0000.

This online quadratic equation solver acts like a specialized function you'd find on a Texas Instruments (TI) graphing calculator, providing quick and accurate solutions for a common mathematical problem. It's a powerful tool for students, educators, and professionals alike.