Problem 27 - College Algebra

Question

Multiply or divide as indicated. $$\frac{4 x^{2}+10}{x-3} \div \frac{6 x^{2}+15}{x^{2}-9}$$

Step-by-Step Solution

Answer

The simplified expression of the problem is \( \frac{2(x+3)}{3(x-3)} \)

Step 1: Write Division as Multiplication

Rewrite the expression \( \frac{4 x^{2}+10}{x-3} \div \frac{6 x^{2}+15}{x^{2}-9} \) as multiplication by following the rule \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \). The new expression becomes \( \frac{4 x^{2}+10}{x-3} \times \frac{x^{2}-9}{6 x^{2}+15} \) .

Step 2: Factorize the Expressions

Factorize the expressions \(4 x^{2}+10\), \(6 x^{2}+15\), and \(x^{2}-9\). After factoring, the expressions become \(2(2x^{2}+5)\), \( 3(2x^{2}+5)\), and \( (x-3)(x+3)\) respectively.

Step 3: Insert Factored Expressions

Replace the original expressions with the factored expressions in the multiplication expression obtained from step 1: \( \frac{2(2x^{2}+5)}{x-3} \times \frac{(x-3)(x+3)}{3(2x^{2}+5)} \) .

Step 4: Simplify Expression

Simplify the expression by cancelling out similar terms from the numerator and denominator: \( \frac{2}{3} \times \frac{x+3}{x-3} \) (since \(2x^{2}+5\) and \(x-3\) are being cancelled out).

Step 5: Write Final Simplified Expression

The simplified expression is \( \frac{2(x+3)}{3(x-3)} \) .