Show all work to solve 3x2 + 5x − 2 = 0.

By the quadratic formula,

\(x = \frac{-5 \pm \sqrt{5^2 - 4(3)(-2)}}{2} = \frac{-5 \pm 7}{2}\)

 

so the roots are -6 and 1.

first, we can do 3x2.

so this would be 6+5x-2=0

Now, we can move all the numbers to the other side by adding or subtracting them on each side

 

6-6+5x-2+2=0-6+2

 

or

 

5x = -4

 

then divide both sides by 5

 

x = -4/5

To factor the left-hand side, we need 2 numbers that sum to 5 and two numbers that multiply to \(3 \times -2 = -6\). The only two numbers that work are -1 and 6. 

 

Now, rewrite this equation to \(3x^2 - x + 6x - 2 = 0\). Just factor \(3x^2 - x \) to \(x(3x - 1)\) and factor \(6x - 2\) to \(2(3x - 1)\). 

 

Adding both of these gives us \((x+2)(3x-1) = 0\). 

 

If we want the product to be 0, x + 2 = 0, or 3x - 1 = 0. 

 

Can you take it from here?