To find the completely factored form of the quadratic expression [tex]\(8x^2 - 50\)[/tex], let's break it down step by step.
First, observe that [tex]\(8x^2 - 50\)[/tex] can be written as:
[tex]\[8x^2 - 50\][/tex]
We start by factoring out the greatest common factor (GCF). The GCF of the terms [tex]\(8x^2\)[/tex] and [tex]\(-50\)[/tex] is 2. So, we factor out 2 from the expression:
[tex]\[8x^2 - 50 = 2(4x^2 - 25)\][/tex]
Next, we focus on factoring the expression inside the parentheses: [tex]\(4x^2 - 25\)[/tex]. Notice that this is a difference of squares. Recall the difference of squares formula:
[tex]\[a^2 - b^2 = (a - b)(a + b)\][/tex]
We can apply this formula to [tex]\(4x^2 - 25\)[/tex], where [tex]\(4x^2\)[/tex] is [tex]\((2x)^2\)[/tex] and [tex]\(25\)[/tex] is [tex]\(5^2\)[/tex]:
[tex]\[4x^2 - 25 = (2x)^2 - 5^2\][/tex]
Using the difference of squares formula, we have:
[tex]\[4x^2 - 25 = (2x - 5)(2x + 5)\][/tex]
Finally, substituting back into our original factored form, we get:
[tex]\[8x^2 - 50 = 2(2x - 5)(2x + 5)\][/tex]
Thus, the completely factored form of [tex]\(8x^2 - 50\)[/tex] is:
[tex]\[2(2x - 5)(2x + 5)\][/tex]
So, the correct answer is:
[tex]\[2(2x+5)(2x-5)\][/tex]
