Problem 118 - Algebra and Trigonometry

Question

Explain how to recognize an equation that is quadratic in form. Provide two original examples with your explanation.

Step-by-Step Solution

Answer

Essentially, an equation is quadratic if it can be arranged in the form ax^2 + bx + c = 0, with the highest power of the variable being 2 and 'a' not equating to 0. An example could be 5x^2 + 3x - 8 = 0 or -2x^2 - 7x + 3 = 0.

Step 1: Understand the structure of a quadratic equation

A quadratic equation, in standard form, is written as ax^2 + bx + c = 0. The power of x stands at 2, making it a second degree polynomial. 'a' should not be equal to 0, where 'a', 'b' and 'c' are constants.

Step 2: Identify the quadratic form in an equation

Upon examining an equation, if you find a variable that is squared, or if the equation can be rearranged into the form ax^2 + bx + c = 0, then the equation is quadratic in form.

Step 3: Example 1 of a quadratic equation

Consider the equation 5x^2 + 3x - 8 = 0. This equation aligns with the aforementioned standard form for it has 'x' raised to the power of 2 which denotes it as being a quadratic equation.

Step 4: Example 2 of a quadratic equation

Consider the equation -2x^2 - 7x + 3 = 0. Similar to Step 3, this equation matches the standard form, constituting it as a quadratic equation as it has 'x' to the power of 2.