Factor completely $25 x^2-36$.
The expression given is \(25x^2 - 36\). To factor this, we analyze its structure. Notice that both terms are perfect squares: \(25x^2 = (5x)^2\) and \(36 = 6^2\). This indicates that the expression is in the form of a difference of two squares, \(A^2 - B^2\).
The difference of squares identity states that for any two algebraic terms \(A\) and \(B\), the expression \(A^2 - B^2\) can be factored as \((A - B)(A + B)\). Here, we have identified \(A = 5x\) and \(B = 6\).
Using the identity identified in the previous step, we substitute \(A = 5x\) and \(B = 6\) into the formula:
\[25x^2 - 36 = (5x)^2 - (6)^2 = (5x - 6)(5x + 6).\]
Comparing this result with the provided choices:
The factored form \((5x - 6)(5x + 6)\) corresponds to the first option.
In engineering and physics, factoring differences of squares is frequently used when simplifying complex equations that model the physical properties of objects. For example, if you are determining the force exerted by an object whose energy is defined by an equation like \(E = 25x^2 - 36\), factoring this into \((5x-6)(5x+6)\) allows you to find the specific values of \(x\) (the roots) where the energy is zero, i.e., \(5x-6=0\) or \(5x+6=0\), simplifying calculations for structural integrity or speed.