Factoring A Difference Between Two Squares Lessons
Take a look at the problem and work below:
x ^{2} - 2x + 3x -6
x ^{2} + x - 6
As you can see above, two binomials in parentheses, (x + 3) and (x - 2) are multiplied using the FOIL method. When the outer terms and inner terms are multiplied they result in -2x and 3x. Because these two resulting terms have different coefficients, when we combine like terms the result is another term, +x, which appears in the last line of the problem.
Now observe the following problem:
x ^{2} + 2x - 2x - 4
x ^{2} - 4
In this case, the multiplication of the outer terms and of the inner terms resulted in 2x and -2x. The terms 2x and -2x have coefficients of 2 and -2. These coefficients are essentially the same number, but with with opposite signs (one number is positive and the other is negative). These two terms form a zero pair, meaning when they are combined, they cancel each other out. You can see that in the last step of the problem, where like terms were combined, the zero pair 2x and -2x canceled out. As a result, only two terms remained on the last line.
The result from the last problem is called a Difference Between Two Squares. A Difference Between Two Squares is an expression with two terms (also known as a binomial) in which both terms are perfect squares and one of the two terms is negative.
The problems that follow show how to factor a difference between two squares. The factoring process, which converts an expression like "x^{2} - 4" into "(x - 2)(x + 2)", is essentially the opposite of the multiplication process we used above.
Factoring A Difference Between Two Squares
Take a look at the problem (expression) below:
The first step at factoring this is to make sure that the expression is a difference between squares.
Question | Answer and Reason |
Are there only two terms? | Yes. The first term is 4x^{2}; the second term is -16. |
Are both coefficients (4 and 16) perfect squares? | Yes. Notice 2 times 2 equals 4, and 4 times 4 equals 16. |
Are all of the variables in the expression raised to an even (2,4,6, ...) power? | Yes. There is only one variable, x, and it has a power of 2 which is even. |
Does one term have a positive coefficient, and another term have a negative coefficient? | Yes. The coefficient 4 is positive, and the coefficient -16 is negative. |
Because "Yes" was answered to each of the above questions, we know that the expression is a difference between two squares. Begin the factoring process by writing two sets of open parentheses:
Now find the square root of 4x^{2}, the first term, by finding the square root of 4 and then dividing each exponent by 2. The square root of 4 is 2. Half of the exponent 2 is 1, thus x^{2} becomes x^{1} or x. Thus, the square root of the entire term is 2x. Write this term on the left inside of each set of parentheses.
We will now consider 16, the second term without the negative sign. We will apply the same process that we applied to 4x^{2}. There are no variables in 16, so we simply find that the square root of 16 is 4. Now 4 is written on the right inside of each set of parentheses.
Add a plus sign to the middle of the first set of parentheses, then add a minus sign to the middle of the second set of parentheses.
The result is two parentheses which can be multiplied to get the original expression 4x^{2} - 16. To check that this answer is correct, you can apply the FOIL Method which was presented in an earlier lesson.
Factoring A Difference Between Two Squares
Examine the problem / expression below:
Again, the first step at factoring this expression is to verify that the expression is a Difference Between Two Squares.
Question | Answer and Reason |
Are there only two terms? | Yes. The first term is -9; the second term is j^{4}. |
Are both coefficients (9 and 1) perfect squares? | Yes. Notice 3 times 3 equals 9 and 1 times 1 equals 1. |
Are all of the variables in the expression raised to an even (2,4,6, ...) power? | Yes. There is only one variable, j, and it has a power of 4 which is even. |
Does one term have a positive coefficient, and another term have a negative coefficient? | Yes. The coefficient 1 is positive, the coefficient -9 is negative. |
Unlike the last problem, the first term is negative. To make factoring this expression easier, simply switch the two terms so that the negative term is second.
becomes
j ^{4} - 9
Now, continue factoring as in the last problem. Write two sets of open parentheses:
Find the square root of the first term, j^{4}. Write the result, j ^{2}, on the left inside of each set of parentheses.
Find the square root of the second term, 9. Write the result, 3, on the right inside of each set of parentheses.
Now write a plus sign in the middle of the first set of parentheses and write a minus sign in the middle of the second set of parentheses.
Difference Between Two Squares Resources
Practice Problems / Worksheet Expression Factoring Calculator
Next Lesson:
Factoring a Trinomial |
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