Problem 24 - Algebra and Trigonometry

Question

Factor each trinomial, or state that the trinomial is prime. $$2 x^{2}+5 x-3$$

Step-by-Step Solution

Answer

The factored form of the trinomial \(2x^2+5x-3\) is \((2x-1)(x+3)\).

Step 1: Identify the coefficients.

The first step is to identify the coefficients from the trinomial. Here, 'a' is 2, 'b' is 5, and 'c' is -3 in the standard form of a quadratic equation \(ax^2+bx+c\).

Step 2: Find two numbers that add up to b and multiply to ac.

Now, search for two numbers that add up to 'b' (which is 5) and multiply to 'ac' (which is \(-2*3=-6\)). The numbers here that satisfy this requirement are 6 and -1 since \(6*-1=-6\) and \(6-1=5\).

Step 3: Rewrite the trinomial.

Rewrite the middle term of the trinomial using the two numbers found. The equation now becomes \(2x^2+6x-x-3\).

Step 4: Factor by grouping.

Now, perform factor by grouping. The first two terms can be grouped together, and the last two terms can be grouped together. This gives \(2x(x+3)-1(x+3)\).

Step 5: Write the final factored form.

Finally, write the final factored form as \((2x - 1)(x+3)\) as the expression \(x+3\) was common between the terms.