Question

Show all work to solve 3x^2 − 5x − 2 = 0
Factor by grouping.

Answer 1

Answer:

x = 2

x = -1/3

Step-by-step explanation:

Here we go!

First, we’ll multiply the coefficient of the first term by the constant: 3 * -2 = -6. Now, we want to find two factors of -6 whose sum equals the coefficient of the middle term (-5). I see that -6 * 1 = -6 and -6 + 1 = -5, so it looks like -6 and 1 satisfy those conditions.

Let’s rewrite the original polynomial, splitting the middle term using the two factors we just found: 3x^2 - 6x + 1x - 2 = 0. I’m going to group this equation into two parts with parenthesis now. This doesn’t change the meaning—it just makes the next step easier to visualize: (3x^2 - 6x) + (1x - 2) = 0.

I’ll pull out a common factor, 3x, from the first part: 3x(x - 2) + (1x - 2) = 0. Now I’ll pull a 1 out of the second part: 3x(x - 2) + 1 (x - 2) = 0. Check it out—the expressions in the parenthesis now match! They’re both (x - 2).

Now our equation can be simplified: (3x+1)(x - 2) = 0. I put together the common factors on the outside, and I combined the two (x-2) groups into one.

You can now set each of the parenthesis groupings equal to 0 and solve each one for x.

Answer 2

Therefore the solution to the equation 3x² - 5x - 2 = 0 is

x = 2 or x = -1/3

The given equation is:

$3x^2 - 5x - 2 = 0$

Multiply the first and last term together

3  x  2  =  6

Other factors of 6 are 1 and 6

Combine 1x and 6x to replace the middle term (-5x)

$3x^2-5x-2=0\\\\3x^2-6x+x-2=0$

Factor 3x out of the first two terms and 1 out of the last two terms

$3x(x - 2)+1(x - 2) = 0\\\\(x-2)(3x+1) = 0$

Set x - 2 to 0

$x-2 = 0\\\\x = 2$

Set 3x+1 to 0

$3x+1 = 0\\\\3x = -1\\\\x = \frac{-1}{3}$

Therefore the solution to the equation 3x² - 5x - 2 = 0 is x = 2 or x = -1/3

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