That is, when we encounter two numbers with a minus sign between them, that is, the difference and they will be in parentheses and raised as a squared expression, we can use this formula.
Pay attention - The formula also works in non-algebraic expressions or combinations with numbers and unknowns.
Let's look at an example
(X−7)2= Here we identify two elements between which there is a minus sign and they are enclosed in parentheses and raised to the square as a single expression. Therefore, we can use the formula for the difference of squares. We will work according to the formula and pay attention to the minus and plus signs. We will obtain: (X−7)2=x2−14x+49 Indeed, we pronounce the same expression differently using the formula.
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Examples and exercises with solutions of the difference of squares formula
Exercise #1
Choose the expression that has the same value as the following:
(x−y)2
Video Solution
Step-by-Step Solution
We use the abbreviated multiplication formula:
(x−y)(x−y)=
x2−xy−yx+y2=
x2−2xy+y2
Answer
x2−2xy+y2
Exercise #2
60−16y+y2=−4
Video Solution
Step-by-Step Solution
Let's solve the given equation:
60−16y+y2=−4First, let's arrange the equation by moving terms:
60−16y+y2=−460−16y+y2+4=0y2−16y+64=0Now, let's note that we can break down the expression on the left side using the short quadratic factoring formula:
(a−b)2=a2−2ab+b2This is done using the fact that:
64=82So let's present the outer term on the right as a square:
y2−16y+64=0↓y2−16y+82=0Now let's examine again the short factoring formula we mentioned earlier:
(a−b)2=a2−2ab+b2And the expression on the left side of the equation we got in the last step:
y2−16y+82=0Let's note that the terms y2,82indeed match the form of the first and third terms in the short multiplication formula (which are highlighted in red and blue),
But in order for us to break down the relevant expression (which is on the left side of the equation) using the short formula we mentioned, the match to the short formula must also apply to the remaining term, meaning the middle term in the expression (underlined):
(a−b)2=a2−2ab+b2In other words - we'll ask if it's possible to present the expression on the left side of the equation as:
y2−16y+82=0↕?y2−2⋅y⋅8+82=0And indeed it holds that:
2⋅y⋅8=16ySo we can present the expression on the left side of the given equation as a difference of two squares:
y2−2⋅y⋅8+82=0↓(y−8)2=0From here we can take out square roots for the two sides of the equation (remember that there are two possibilities - positive and negative when taking out square roots), we'll solve it easily by isolating the variable on one side:
(y−8)2=0/y−8=±0y−8=0y=8
Let's summarize then the solution of the equation: