Problem 78 - Algebra and Trigonometry

Question

Exercises \(78-80\) will help you prepare for the material covered in the next section. Factor: \(2 x^{2}+7 x-4\)

Step-by-Step Solution

Answer

The factored form of the quadratic equation \(2x^{2}+7x-4\) is \(2(x+2)(x-2)\).

Step 1: Identifying the Coefficients

Identify the coefficients for the quadratic equation. In this case, a = 2, b = 7, and c = -4.

Step 2: Finding two numbers

Find two numbers such that they add up to b (which is 7) and their product is equal to the product of a and c (which is -8). The numbers in this case are 4 and -2.

Step 3: Split the Middle Term

Split the middle term of the quadratic equation using the two numbers found above. It then becomes: \(2x^{2}+4x-2x-4\).

Step 4: Group the Terms

Group the terms and factor by grouping. This will result in \(2x(x+2)-2(x+2)\).

Step 5: Factoring

Finally, we can factor out the common binomial term to obtain the factorized form of the quadratic equation. The factors are: \(2(x+2)(x-2)\).