Problem 85 - College Algebra Essentials

Question

Perform the indicated operation or operations. $$(5 x-7)(3 x-2)-(4 x-5)(6 x-1)$$

Step-by-Step Solution

Answer

\(-9x^2 + 3x + 9\)

Step 1: Multiply the Binomials

Applying the FOIL method (First, Outer, Inner, Last), multiply the binomials. The operation is expanded as follows. First pair: \(5x * 3x = 15x^2\)Outer pair: \(5x * -2 = -10x\)Inner pair: \(-7 * 3x = -21x\)Last pair: \(-7 * -2 =14\)Add all these together to get \(15x^2 - 31x + 14\). Similarly for the second binomial, we getFirst pair: \(4x * 6x = 24x^2\)Outer pair: \(4x * -1 = -4x\)Inner pair: \(-5 * 6x = -30x\)Last pair \(-5 * -1 = 5\)Add these together to get: \(24x^2 - 34x + 5\)

Step 2: Subtract the Multiplications

Subtract the second multiplication result from the first: This is done by subtracting each corresponding term in the two expressions:\((15x^2 - 31x + 14) - (24x^2 - 34x + 5)\).Resulting in:\(-9x^2 + 3x + 9\)

Step 3: Simplify

There aren't any like terms remaining, so the final simplified expression is :\(-9x^2 + 3x + 9\)