Difference Of Perfect Squares

[Annemie]

Iftwo terms in a binomial are perfect squares separated by subtraction, then you can factor them. To factor the difference of two perfect squares, remember this rule: if subtraction separates two squared terms, then the sum and the difference of the two square roots factor the binomial. For example:

Students sometimes begin problems like these by making a list. The list might start out rather haphazardly, but often when we organize a list, we can make some useful observations. For example, suppose that we made a list like the following:

We can also represent the odd numbers, as well as other numbers, as the differences of perfect squares visually on grids. In each case below, a shaded perfect square grid is cut out from the corner of a larger perfect square grid, leaving a shape that represents the difference of two perfect square whole numbers.

Thus, only even numbers that are multiples of 4 can be expressed as the difference of two perfect square whole numbers. Now the question is, Given any number that is a multiple of 4, could we in fact write it as the difference of two perfect squares?

One way to factor an expression is to use the difference of two squares. Writing a binomial as the difference of two squares simply means you rewrite a binomial as the product of two sets of parentheses multiplied by each other. For example, \(a^2-b^2=(a+b)(a-b)\). The binomial \(a^2-b^2\) can be factored into two sets of parentheses multiplied by each other. \((a+b)(a-b)\) will produce \(a^2-b^2\) when multiplied.

Not all expressions can be factored using this method. There are a few clues to look for when determining whether an expression can be factored using the difference of squares. Notice in the previous example \(a^2-b^2\) that each term is a perfect square, and there is a subtraction symbol between each term. These are two helpful clues to look for when determining if a binomial can be factored using the difference of two squares. If these two clues are present, then the expression can be factored using the difference of squares.

The original subtraction symbol between the terms is what allows these middle terms to cancel out. If the original expression was \(x^2+9\), with addition between the two terms, it would not be possible to factor this using the difference of squares. If we tried to break this apart into \((x+3)(x+3)\), the result would be \(x^2+3x+3x+9\), which simplifies to \(x^2+6x+9\), which does not match the original expression. The important thing to remember is that subtraction between the two terms is required for an expression to be factored using the difference of squares.

Consider the binomial \(4x^2-49\). Can this be factored using the difference of squares? Subtraction is the symbol between both terms, which is a good start. The first term is raised to the power of two which is also good. When we look at the second term, \(49\), we notice that this can be written as \(72\). Now we have \(4x^2-7^2\). All clues indicate that the expression \(4x^2-49\) can be rewritten as the difference of two squares. \(4x^2-49\) becomes \((2x+7)(2x-7)\). We can check this by (FOIL)ing and checking that the product is in fact \(4x^2-49\).

To review, we know that a binomial can be written as the difference of two squares if both terms are perfect squares and they are separated by subtraction. The difference of two squares is a useful theorem because it tells us if a quadratic equation can be written as the product of two binomials.

The expression \(9x^2-16y^2\) can be factored using the difference of two squares because both terms are perfect squares, and the squared terms are separated by the subtraction symbol. \(9x^2\) can be expressed as \(3x\) times \(3x\), and \(16y^2\) can be expressed as \(4y\) times \(4y\). \(9x^2-16y^2\) becomes \((3x+4y)(3x-4y)\).

\(x^2-2\) is separated by subtraction, and the first term is a perfect square, however \(2\) is not a perfect square. There is no number that can be squared, with a result of \(2\). This expression cannot be factored using the difference of squares.

If both terms in a binomial are perfect squares, and the terms are separated by subtraction, then the binomial can be factored using the difference of squares. For example, \(x^2-4\) can be factored using the difference of squares because both terms are perfect squares, and they are separated by a subtraction sign.

The binomial \(9x^2-49y^2\) can be factored using the difference of squares because both terms are perfect squares and they are separated by a subtraction symbol. The first term \(9x^2\) can be split into \(3x\) times \(3x\). The second term \(49y^2\) can be split into \(7y\) times \(7y\). One set of parentheses needs to be addition and the other needs to be subtraction so that the middle term cancels out when FOILing.

(\(21xy\) and \(-21xy\) cancel out)

\(9x^2-49y^2\) becomes \((3x+7y)(3x-7y)\).

This is a factoring calculator if specifically for the factorization of the difference of two squares. If the input equation can be put in the form of a2 - b2 it will be factored. The work for the solution will be shown for factoring out any greatest common factors then calculating a difference of 2 squares using the idenity:

If a is negative and we have addition such that we have -a2 + b2 the equation can be rearranged to the form of b2 - a2which is the correct equation only the letters a and b are switched; we can just rename our terms.

Whenever you have a binomial with each term being squared (having an exponent of [latex]2[/latex]), and they have subtraction as the middle sign, you are guaranteed to have the case of difference of two squares.

These are other ways to write the formula of the difference of two squares using variables. Learn to recognize them in various appearances so that you know exactly how to handle them.

The first term of the binomial is definitely a perfect square because the variable [latex]x[/latex] is being raised to the second power. However, the second term of the binomial is not written as a square. So we need to rewrite it in such a way that [latex]9[/latex] is expressed as some number with a power of [latex]2[/latex]. I hope you can see that [latex]9 = \left( 3 \right)^2[/latex]. Clearly, we have a difference of two squares because the sign between the two squared terms is subtraction.

For this example, the solution is broken down in just a few steps to highlight the procedure. Once you get comfortable with the process, you can skip a lot of steps. In fact, you can go straight from the difference of two squares to its factors.

Now, we can truly rewrite this binomial as the difference of two squares with distinct terms that are being raised to the second power; where [latex]16y^4 = \left( 4y^2 \right)^2[/latex] and [latex]81 = \left( 9 \right)^2[/latex]

Are we done already? Well, examine carefully the binomials you factored out. The second parenthesis is possibly a case of difference of two squares as well since [latex]4y^2 = \left( 2y \right)\left( 2y \right)[/latex] and clearly, [latex]9 = \left( 3 \right)\left( 3 \right)[/latex].

Now we can deal with the binomial inside the parenthesis. It is actually a difference of two squares because we can express each term of the binomial as an expression with a power of [latex]2[/latex].

You may keep it in that form as your final answer. But the best answer is to combine like terms by adding or subtracting the constants. This also simplifies the answer by getting rid of the inner parenthesis.

Not sure if the binomial you've factoring is a difference of squares problem? This tutorial will show you what characteristics the binomial must have in order to be a difference of squares problem. Take a look!

Anytime you square an integer, the result is a perfect square! The numbers 4, 9, 16, and 25 are just a few perfect squares, but there are infinitely more! Check out this tutorial, and then see if you can find some more perfect squares!

Monomials are just math expressions with a bunch of numbers and variables multiplied together, and one way to compare monomials is to keep track of the degree. So what's a degree? Well, if you've ever wondered what 'degree' means, then this is the tutorial for you.

There is a bunch of vocabulary that you just need to know when it comes to algebra, and coefficient is one of the key words that you have to feel 100% comfortable with. Check out the tutorial and let us know if you want to learn more about coefficients!

Polynomials are those expressions that have variables raised to all sorts of powers and multiplied by all types of numbers. When you work with polynomials you need to know a bit of vocabulary, and one of the words you need to feel comfortable with is 'term'. So check out this tutorial, where you'll learn exactly what a 'term' in a polynomial is all about.

In mathematics, the difference of two squares is a squared (multiplied by itself) number subtracted from another squared number. Every difference of squares may be factored according to the identity

Since the two factors found by this method are complex conjugates, we can use this in reverse as a method of multiplying a complex number to get a real number. This is used to get real denominators in complex fractions.[1]

The difference of two squares can also be used in the rationalising of irrational denominators.[2] This is a method for removing surds from expressions (or at least moving them), applying to division by some combinations involving square roots.

The difference of two squares can also be used as an arithmetical short cut. If two numbers (whose average is a number which is easily squared) are multiplied, the difference of two squares can be used to give you the product of the original two numbers.

A ramification of the difference of consecutive squares, Galileo's law of odd numbers states that the distance covered by an object falling without resistance in uniform gravity in successive equal time intervals is linearly proportional to the odd numbers. That is, if a body falling from rest covers a certain distance during an arbitrary time interval, it will cover 3, 5, 7, etc. times that distance in the subsequent time intervals of the same length.

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