Order of operations in algebra

When creating simpler and more useful expressions, you want to be careful not to change the original value. By applying the order of operations, you maintain that value.

Apply the order of operations when no grouping symbols, such as parentheses, interrupt. When more of one level occurs in a problem, do them in order from left to right. When you perform operations on algebraic expressions and you have a choice between one or more operations to perform, use the following order:

1. Powers and roots

2. Multiplication and division

3. Addition and subtraction

These rules are interrupted if the problem has grouping symbols. You first need to perform operations in grouping symbols, such as ( ), { }, [ ] , above and below fraction lines, and inside radicals.

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Rules of exponents

Exponents are shorthand for repeated multiplication. The rules for performing operations involving exponents allow you to change multiplication and division expressions with the same base to something simpler. Remember that in xa the x is the base and the “a�? is the exponent.

Assume that neither x nor y are equal to zero:

Selected math formulas step by step

Algebraic formulas make life (and algebra) simpler. You save time by not having to perform more complicated tasks. When using the formulas, use the appropriate rules for simplifying algebraic expressions. Also watch out for pitfalls; to help you, an asterisk (*) appears beside steps where errors are easy to make.

QUADRATIC FORMULA EXAMPLES

Definition: A quadratic equation is a function of the form ax² + bx + c = 0 (where a does not equal zero). On a graph, a quadratic equation can be represented by a parabola. The x-values where the parabola crosses the x-axis is called the solutions, or roots, of the quadratic equation.

For example, consider the following quadratic equation:

x² + 5x + 6 = 0

Notice that this equation is in ax² + bx + c = 0 form, where…

a=1

b=5

c=6

If we want to find the solutions, or roots, of this quadratic equation, we have a few options.

First, we could factor this quadratic equation by looking for two values that add to 5 and also multiply to 6, which, in this case, would be 2 and 3. So we could say that…

• x² + 5x + 6 = 0 → (x+2)(x+3) = 0

We could then solve for each factor as follows:

• x + 2 = 0 → x = -2

• x + 3 = 0 → x = -3

Now we can conclude that the solutions of this quadratic are x=-2 or x=-3.

Another option for finding the solutions to a quadratic equation is to look at its graph. The solutions, or roots, will be the x-values where the graph crosses the x—axis. Note that quadratic equations can have two roots, one root, or even no real roots (as you will see later in this guide).

As for the equation x² + 5x + 6 = 0, the corresponding graph above confirms that the equation has solutions at x=-2 and x=-3.

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But what do we do when a quadratic equation is very difficult to factor or when we do not have access to a clear graph? Well, this is where the quadratic formula comes into play.

the quadratic formula 

Definition: Any quadratic equation of the form ax² + bx + c = 0 (where a ≠ 0), can be solved using the quadratic formula, which states that…

• x= (-b ± [√(b² - 4ac)]) / 2a

Why is the quadratic formula so useful? Because, as the definition states, it can be used to find the solutions to any quadratic equation. While the quadratic equation that we just looked at, x² + 5x + 6 = 0, was pretty easy to work with and solve, it is considered extremely simple. As you move farther along your algebra journey, you will come across more and more complex quadratic equations that can be very difficult to factor or even graph.

However, if you know how to use the quadratic formula, you can successfully solve any quadratic equation. With this in mind, let’s go ahead and work through some quadratic formula examples to gain some practice.

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 quadratic formula example 1

And we will start by using it to solve x² + 5x + 6 = 0, because we already know that the solutions are x=-2 and x=-3.  If the quadratic formula works, then it should  yield us that same result. Once we work through this first simple example, we will move onto more complex examples of how to use the quadratic formula to solve quadratic equations. 

To use the quadratic formula, start by identifying the values of a, b, and c:

Quadratic Formula Examples

Example #1: 

Solve:  x² + 5x + 6 = 0

First, notice that our equation is in ax² + bx + c = 0 form where:

• a=1

• b=5

• c=6

Identifying the values of a, b, and c will always be the first step (provided that the equation is already in ax² + bx + c = 0 form).

Now that we know the values of a, b, and c, we can plug them into the quadratic equation as follows:

• x= (-b ± [√(b² - 4ac)]) / 2a

• x= -(5) ± [√(5² - 4(1)(6))]) / 2(1)

• x= -5 ± [√(25 - 24)] / 2

• x= -5 ± [√(1)] / 2

• x= (-5 ± 1) / 2

Now we are left with x= (-5 ± 1) / 2. Note that the ± mean “plus or minus” meaning that we have to split this result into two separate equations:

• Plus: x = (-5 + 1) / 2

• Minus: x = (-5 - 1) / 2

By solving these two separate equations, we can find the solutions to the quadratic function x² + 5x + 6 = 0.

• x = (-5 + 1) / 2 = -4/2 = -2 → x=-2

• x = (-5 - 1) / 2 = -6/2 = -3 → x=-3

After solving both equations, we are left with x=-2 and x=-3, which we already knew were the solutions to x² + 5x + 6 = 0. So, we have confirmed that the quadratic formula can be used to find the solutions to any quadratic equation of the form ax² + bx + c = 0.

Final Answer: x=-2 and x=-3

The steps to solving the quadratic formula example is illustrated below: