Solving Equations

The quadratic formula is used to find the solutions of a quadratic equation in the form $$ax^2 + bx + c = 0$$ . It is given by $$x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}$$ .

To solve an equation with absolute values, you set up two cases: one for the positive scenario and one for the negative scenario. For example, if $$|x| = a$$ , then $$x = a$$ or $$x = -a$$ .

To isolate a variable, perform inverse operations to both sides of the equation. This includes adding, subtracting, multiplying, or dividing by the same number, ensuring to maintain the equality.

Writing an answer in simplest form means expressing it in the most reduced or compact way, such as reducing fractions or simplifying radicals.

The discriminant, given by $$b^2 - 4ac$$ , determines the nature of the roots of a quadratic equation. If it is positive, there are two distinct real roots; if zero, one real root; and if negative, two complex roots.

To convert Celsius to Fahrenheit, use the formula $$F = \frac{9}{5}C + 32$$ .

The first step in solving a linear equation is to simplify both sides of the equation, which may include distributing, combining like terms, and eliminating fractions.

Set notation is a mathematical way to represent a collection of objects, typically written in curly braces. For example, the set of solutions can be represented as $$\{x_1, x_2\}$$ .

To solve for a variable in a proportion, cross-multiply and then isolate the variable. For example, if $$\frac{a}{b} = \frac{c}{d}$$ , then $$a \cdot d = b \cdot c$$ .

An equation states that two expressions are equal, typically containing an equals sign (e.g., $$x + 2 = 5$$ ), while an expression is a combination of numbers, variables, and operations without an equals sign (e.g., $$x + 2$$ ).