Mathematics

Algebraic Skills

# Quadratic Equations

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Quadratic Equations

Quadratic Equations

Understanding Quadratic Equations

- A
**quadratic equation**is a second order polynomial equation in a single variable x, with a non-zero coefficient for x². - It has the general form
**ax² + bx + c = 0**, where a, b, and c are constants also known as coefficients, and**a ≠ 0**. - The term
**quadratic**comes from "quadratum," the Latin word for square.

Basics of Quadratic Equations

- The highest power in a quadratic equation is always
**2**. - Quadratic equations can have either
**two distinct**,**one**, or**no real solution**at all, these are determined by the discriminant value. - The graph of a quadratic equation is a
**parabola**. If a > 0, the parabola opens upwards, and if a < 0, it opens downwards.

Solving Quadratic Equations

- Quadratic equations can be solved by three main methods:
**factoring**, using the**quadratic formula**, or**completing the square**. - The quadratic formula is
**x = [-b ± sqrt(b² - 4ac)] / (2a)**. Here the term under the square root,**b² - 4ac**, is called the discriminant. - A quadratic equation can be solved by
**factoring**if it can be expressed in the form of (px + q)(rx + s) = 0.

Discriminant Value

- The
**discriminant**can identify the nature of the roots of the quadratic equation. - If the
**discriminant > 0**, the quadratic equation has**two distinct real solutions**. - If the
**discriminant = 0**, the quadratic equation has**one real solution**, also known as a repeated root. - If the
**discriminant < 0**, the quadratic equation has**no real solutions**but two complex solutions.

Quadratic Roots and Coefficients

- The
**sum of the roots**of a quadratic equation is equal to**-b/a**and the**product of the roots**is equal to**c/a**. This is true only for equations in standard form ax² + bx + c = 0, and comes from Viète's formulas.

Follow these guidelines when studying quadratic equations and their properties. Practice a range of problems using each method of solution for a thorough understanding.