What is a Quadratic Equation? – Definition, Standard Form, and Applications

1. Introduction In this lesson, you will learn the fundamental definition of a quadratic equation, how to identify its standard form, and explore its real-world applications in physics, finance, and engineering.

2. Core Concept Explanation A quadratic equation is a polynomial equation of the second degree. The term "quadratic" comes from the Latin word "quadratus" for square, meaning the variable is squared. The standard form of a quadratic equation is: ax^2 + bx + c = 0

Where: * x is the unknown variable. * a, b, and c are numerical coefficients (constants). * a \neq 0 (If a = 0, the equation becomes linear, not quadratic).

3. Step-by-Step Method To work with a quadratic equation: 1. Move all terms to one side of the equation so the other side is equal to 0. 2. Combine like terms. 3. Arrange the terms in descending order of their exponents: x^2 term first, x term second, and the constant last. 4. Identify the coefficients a, b, and c.

4. Worked Examples * Example 1: Identify a, b, and c in 3x^2 - 5x + 2 = 0. * Solution: a = 3, b = -5, c = 2. * Example 2: Rewrite x(x - 4) = 5 in standard form and identify the coefficients. * Solution: Expand: x^2 - 4x = 5. Subtract 5: x^2 - 4x - 5 = 0. Here, a = 1, b = -4, c = -5. * Example 3: Identify the coefficients in 7 - 2x^2 = 0. * Solution: Rearrange: -2x^2 + 0x + 7 = 0. Here, a = -2, b = 0, c = 7.

5. Exam Relevance * Example WAEC Question: Express (2x - 3)(x + 1) = 4 in standard form. * Solution: Expand: 2x^2 + 2x - 3x - 3 = 4 \Rightarrow 2x^2 - x - 3 = 4. Subtract 4: 2x^2 - x - 7 = 0.

6. Common Mistakes * Sign Errors: Forgetting that signs belong to the coefficients. In x^2 - 3x - 4 = 0, b = -3 and c = -4, not 3 and 4. * Missing Standard Form: Attempting to solve before setting the equation to zero. Always set to 0 first.

7. Practice Questions 1. Write (x-2)^2 = 3x + 1 in standard form. * Answer: x^2 - 7x + 3 = 0. 2. Identify coefficients in x^2/2 - x + 4 = 0. * Answer: a = 0.5, b = -1, c = 4.

8. Summary / Cheat Sheet * Standard form: ax^2 + bx + c = 0. * a is the coefficient of x^2, b is the coefficient of x, and c is the constant. * a can never be zero.