When you solve a quadratic equation, you are finding the values of the variable (usually x) that make the equation true. These values are also called roots or zeros of the quadratic equation.
Understanding Quadratic Equations
A quadratic equation is generally expressed in the form:
ax² + bx + c = 0Where:
- a, b, and c are known as the quadratic coefficients. a cannot be zero.
What You're Actually Finding
According to the reference provided, the solutions to a quadratic equation are the values of the unknown variable x, which satisfy the equation. These solutions are known as roots or zeros of the quadratic equation. The roots represent the x-intercepts of the parabola defined by the quadratic equation when graphed.
Methods for Solving Quadratic Equations
Several methods can be used to find these roots, including:
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Factoring: Breaking down the quadratic expression into a product of two linear expressions.
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Completing the Square: Transforming the equation into a perfect square trinomial.
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Quadratic Formula: A general formula that provides the solutions for any quadratic equation:
x = (-b ± √(b² - 4ac)) / (2a)
Example
Consider the quadratic equation:
x² - 5x + 6 = 0Solving this equation (e.g., by factoring) gives us the solutions:
- x = 2
- x = 3
These values, 2 and 3, are the roots or zeros of the equation, meaning that if you substitute either of these values for x in the original equation, the equation will be true.
Roots and Zeros
The terms "roots" and "zeros" are often used interchangeably. They represent the values of x where the quadratic equation equals zero. When graphed, these values correspond to the points where the parabola intersects the x-axis.
