Higher Mathematics SQA

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.