Algebra forms the foundation of all the higher forms of math. For that reason, it is imperative that you feel comfortable working with its many forms. Many professions outside mathematics use algebra on a regular basis such as business professionals, physicists, engineers and computer programmers. This How to Solve Algebra Equations tutorial will demonstrate how to handle several different types of equations.

You will need to have a strong grasp of basic math skills such as working with fractions, exponents and signs in order to follow along easily. In this how to, you will learn how to solve for one variable, multiple variables and quadratic equations using various common methods. Although it isn't required, you may want to plot the points from the quadratic equation onto graph paper for your own visual information.

This is the simplest form in algebra. You should have no problems with it as long as you pay attention to the signs and bring it to its simplest form on both sides before beginning.

1. Simplify both sides of the equal signs.
2. Use subtraction or addition to get the variable and its coefficient(number in front of the variable) alone on one side by itself. (Whatever you do to one side you must do to the other.)
3. Once again, check if any terms need to be simplified.
4. Use division or multiplication to remove the coefficient by doing it to both sides of the equation.
5. Check your answer by placing x back into the equation.

At this point, the equation may be a little more complex. However, it is still just a matter of proceeding methodically through each step. (Note: The x and y solution that you get is the point where the lines made by these equations intercept each other in space.)

• Method 1: Linear Combination

1. Pick either the x or y coefficient to remove. Use a common multiple between the coefficients of the variable you decide cancel out.
2. Then, subtract one equation from the other.
3. Follow the same procedures for a single variable equation to get x or y by itself on the left.
4. Plug this new equation into either equation.
5. Now, that you should have a numerical answer for one of your variables.
6. Use that number in one of the equations and calculate the other one.
7. Check your answer by placing x and y into one of the equations.

• Method 2: Substitution

1. Choose one of the equations and solve for either x or y.
2. Plug this new equation into either equation in the matching variable location.
3. Simplify both sides of the equal sign.
4. You should have a numerical solution for one of your variables now.
5. Use that number in one of the equations and calculate the other one.
6. Check your answer by placing x and y into one of the equations.

Some people become intimidated by the thought of quadratics. Yet, they aren't that hard to calculate if you learn how to work with the different techniques of solving them. (Note: The graph of a quadratic is a parabola. Therefore, the two x values that come out in the answer are the x intercepts, where the parabola crosses the x axis on a graph.)

• Standard Quadratic Form: Ax² + Bx + C = 0

• Method 1: Factoring

1. Make sure the equation is in standard form.
2. Start by making two sets of parentheses with x in both of them.
3. Find the factors of C. (Find every factor by dividing by prime numbers until only primes are left. Then, multiply the primes together in every possible combination to get all the factors.)
4. Pick the two factors that when added or subtracted give you B.
5. Place one in each parenthesis.
6. Now, make each parenthesis equal to 0 and solve for x.
7. Check your answer with both x values.

Although factoring is the easiest technique, all quadratics can't be solved this way. In this case, you will have to use one of the following solutions.

• Method 2: Completing the Square

1. Move C to the opposite side of A and B.
2. Multiply or divide each term by the coefficient of x². Your goal is to be left with a coefficient of 1 for x².
3. Now, you will need to take half of B and square it.
4. Add this number to both sides.
5. Find the square of the left side (using factoring) and simplify the numerical one.
6. Take the square root of both sides.
7. Solve for the negative and positive solutions.
8. Check your answer with both x values.

• Method 3: Quadratic Formula

1. Make sure the equation is in standard form.
2. Record A, B, and C. This step helps decrease sign errors when you place them in the formula later.
3. Now, plug the values into the formula.
4. Combine all the values down until you reach your x solutions.
5. Check your answer with both x values.

Solve x in x² - 7x +12 = 0

• Start by making two sets of parentheses.
• C is 12 so the factors are 1, 2, 3, 4, 6 and 12.
• 3 and 4 can be added to get 12.
• Add them to the parentheses. Because of the sign rules we now that we need a negative before both numbers to obtain a negative 7 and positive 12.
• Set x - 3 = 0 and y - 3 = 0.
• Solve both and you get x = 3 and x = 4.
• Check your answer with both x values.

Solve x in 5x₂ - 6x -8 = 0

• Move 8 to the opposite side by adding it to both sides.
• Divide each term by 5 to cancel out the x² coefficient.
• Now, you will need to take half of 6/5 and square it. [(6/5 * 1/2)² = (6/10)² = (3/5)² = 9/25
• Add 9/25 to both sides.
• Find the square of the x² -6/5x - 9/25 (using factoring) and add 8/5 + 9/25 on the other side.
• Take the square root of both sides.
• Solve for the negative and positive solutions.
• Check your answer with both x values.

Solve x in 2x² - x = 1

• A = 2; B = -1; C = -1
• Plug the values into the formula.
• Multiply 4, 2 and -1 in the numerator and 2 times 2 in the denominator.
• Subtract 1 and -8. (Note: The negative gets cancelled out.)
• Take the square root of 9.
• Solve for the negative and positive solutions.
• Check your answer with both x values.