Problem 63 - Beginning and Intermediate Algebra

Question

Factor completely. $$2 d^{2}+2 d-40$$

Step-by-Step Solution

Answer

The completely factored form of \(2d^2 + 2d - 40\) is \(2(d + 5)(d- 4)\).

Step 1: Identify Common Factors

First, let's identify the common factors of the coefficients. In the quadratic expression \(2d^2 + 2d - 40\), the common factor of all terms is 2. So, we can factor out this common factor: \(2(d^2 + d - 20)\) Now we will focus on factoring the quadratic expression inside the parentheses.

Step 2: Identify Factors of Quadratic Expression

We have the quadratic expression \(d^2 + d - 20\). Recall that we need to find two numbers that multiply to -20 and add to 1. After some trial and error, we can find that these two numbers are 5 and -4, because: 5 × (-4) = -20 5 + (-4) = 1

Step 3: Rewrite Quadratic Expression as Product of Binomials

Now that we have found the two numbers (5 and -4), we can rewrite the quadratic expression as a product of two binomials: \(d^2 + d - 20 = (d + 5)(d - 4)\)

Step 4: Combine Factored Expressions

Finally, we will combine the common factor and factored quadratic expression to write the completely factored expression: \(2d^2 + 2d - 40 = 2(d + 5)(d - 4)\) So, the completely factored form of the given quadratic is \(2(d + 5)(d- 4)\).