The Difference of Squares is a way of showing how an expression in the form \(x^2 - y^2\) can be written as the product of 2 binomials, with one as the Sum of Square Roots and the other as the Difference of Square Roots.
It can be expressed as:
Outlined below are the steps required to factor an expression written as a difference of squares:
Factor the binomial \(4x^2 - 25\)
First, as there's no common factor, we can skip Step 1.
Next, we can rewrite the expression in the form \((x)^2 - (y)^2\):
\(= \sqrt{4x^2} - \sqrt{25}\)
\(= 2x - 5\)
\(= (2x)^2 - (5)^2\)
Then, we can use the formula \((x + y)(x - y)\) to factor the rest of the expression:
Therefore, we can determine that \(4x^2 - 25\) expressed as a Difference of Squares is \(\boldsymbol{(2x + 5)(2x - 5)}\).
\(121x^2 - 9y^2\)
First, as there's no common factor, we can skip Step 1.
Next, we can rewrite the expression in the form \((x)^2 - (y)^2\) by taking the square root of each term then raising them to the power \(2\):
\(= \sqrt{121x^2} - \sqrt{9y^2}\)
\(= 11x - 3y\)
\(= (11x)^2 - (3y)^2\)
Then, we can use the formula \((x + y)(x - y)\) to factor the rest of the expression:
Therefore, we can determine that \(121x^2 - 9y^2\) factored is \(\boldsymbol{(11x + 3y)(11x - 3y)}\).
\(18x^2 - 32\)
First, we can factor out the GCF (in this case, \(2\)):
Next, we can rewrite the expression in the form \((x)^2 - (y)^2\):
\(= 2(\sqrt{9x^2} - \sqrt{16})\)
\(= 2(3x - 4)\)
\(= 2((3x)^2 - (4)^2)\)
Then, we can use the formula \((x + y)(x - y)\) to factor the rest of the expression:
Therfore, we can determine that \(18x^2 - 32\) factored is \(\boldsymbol{2(3x + 4)(3x - 4)}\).
\((5c + 3)^2 - (2c + 1)^2\)
First, as there's no common factor, we can skip Step 1. Likewise, as both terms are already squared, we can skip Step 2.
Next, we can use the formula \((x + y)(x - y)\) to factor and simplify the rest of the expression:
\(= (5c + 3) + (2c + 1))((5c + 3 - (2c + 1))\)
\(= (7c + 4)(3c + 2)\)
Therefore, we can determine that \((5c + 3)^2 - (2c + 1)^2\) factored is \(\boldsymbol{(7c + 4)(3c + 2)}\).
First, we can find the area of the shaded area by finding the difference between the area of the whole shape and the area of the white area. This can be expressed as a Difference of Squares:
Next, we can expand and simplify the expression as such:
\(A = [(3x + 4) + (x - 2)][(3x + 4) - (x - 2)]\)
\(A = (4x - 2)(3x + 4 - x + 2)\)
\(A = (4x - 2)(2x + 6)\)
Finally, we can remove out the GCF from both binomials to further simplify the expression:
\(A = 2(2x - 1)2(x - 3)\)
\(A = 4(2x - 3)(x - 1)\)
Therefore, we can determine that the area of the shaded region can be expressed as \(\boldsymbol{4(2x - 3)(x - 1)}\).