# Page 4 Mathematics Study Guide for the ACT

#### Operations in Algebra

The operations in algebra involve solving equalities and inequalities, working with exponents and roots, factoring and simplifying, solving polynomials, linear and quadratic equations, and dealing with functions, word problems and many other concepts.

#### Factoring to Solve Quadratic Equations

The standard form for quadratic equations is:

$ax^2 + bx + c = 0$ where $a \neq 0$

One way to solve or find the roots (or zeroes) of a quadratic equation is by factoring.

Factor $x^2 - 9x + 20 = 0$

In the equation and referring to the standard form, $a = 1$, $b = -9$, and $c = 20$.

What pair of numbers will give you $a \cdot c$ when multiplied together, and give you $b$ when added? $a \cdot c = 20$ and $b = -9$. By intelligent guesswork and lots of practice, you will get the numbers -4 and -5, which when multiplied give 20 and when added give -9.

These numbers give you the factors: $(x - 4)$ and $(x - 5)$. Equating them to zero, you find the roots or zeroes of the equation to be $x = 4$ and $x = 5$.

### Intermediate Algebra

Some quadratic equations are not as easily solved by factoring, and x can be solved using the quadratic formula which is given as:

Solving quadratic equations will yield any of these results:

2 solutions - when the value of $b^2-4ac$ (called the discriminant) is positive; the graph (which is a parabola) touches the x-axis at two points.

1 solution - when the discriminant is equal to zero; the graph touches the x-axis once (at the vertex of the parabola).

2 complex solutions (no real solutions) - when the discriminant is negative; the graph does not touch the x-axis.

Rational expressions are ratios of polynomials. Think of rational expressions as fractions with polynomials in either or both the numerator and denominator. The following are rational expressions:

There is often a need to simplify rational expressions, and factoring is a very useful technique. The second example, for instance, can easily be simplified by factoring:

The factor of $(x+3)$ is common in both the numerator and denominator, and cancels out, leaving the simplified rational expression of:

Polynomials with terms raised to fractional exponents are radical expressions. Knowing how to work with the radicals makes it easier to solve most equations of this type.

To get rid of a square root, cube root or nth root of a term, simply square, cube, or raise it to the nth power. In equations, keep the balance by raising the terms on both sides of the equation to the appropriate power.

Examples:

Remove the radical in the given equation:

Find x in the equation given below:

When you come across expressions with radical denominators, there will be a need to rationalize denominators containing radicals to arrive at the simplest form of your answer. To do that, multiply the numerator and the denominator with the conjugate of the radical expression. Let’s try an example:

Rationalize: $\frac{7}{3-\sqrt{2}}$

#### Absolute Value Equations and Inequalities

Absolute values in equations:

Specific values for the unknown variable in an equation with absolute values can be solved using this property:

$\vert u \vert = a$ is the same as $u = \pm a$

Example:

$x = \frac{22}{5}$ and $x = \frac{6}{5}$

Absolute values in inequalities:

The interval for the unknown variable in an inequality with absolute values can be solved using the following properties:

$\vert x \vert \lt a$ is the same as $-a \lt x \lt a$

$\vert x \vert \le a$ is the same as $-a \le x \le a$

$\vert x \vert \gt a$ is the same as $x \lt -a$ or $x \gt a$

$\vert x \vert \ge a$ is the same as $x \le -a$ or $x \ge a$

Example: