Problem 30 - Beginning and Intermediate Algebra

Question

Factor by grouping. $$6 a^{2}+a b-5 b^{2}$$

Step-by-Step Solution

Answer

The factored expression by grouping for \(6a^2 + ab - 5b^2\) is \((6a + b)(a - 5b)\).

Step 1: Rearrange terms

Rearrange the terms in the expression such that it is easy to group them. In this case, as the terms are already in a suitable order, we don't need to rearrange them. \(6a^2 + ab - 5b^2\)

Step 2: Group the terms in pairs

Now, let's group the terms in pairs that make it easy to factor the common factors. In this case, it is suitable to group two terms each. \((6a^2 + ab) + (-5b^2)\)

Step 3: Factor out the common factor from each group

We can factor out the common factor from each group. In this case, we can factor out "a" from the first group and "-5b" from the second group. \(a(6a + b) - 5b(6a + b)\)

Step 4: Factor the resulting expression

Now, we can factor the common term \((6a + b)\) from the expression. \((6a + b)(a - 5b)\) The expression \(6a^2 + ab - 5b^2\) is now factored by grouping as \((6a + b)(a - 5b)\).