Find the Roots (Zeros) 6x^2-5x=3x^2+2

$6 x^{2} - 5 x = 3 x^{2} + 2$

Step 1

Move all terms containing $x$ to the left side of the equation.

Subtract $3 x^{2}$ from both sides of the equation.

$6 x^{2} - 5 x - 3 x^{2} = 2$

Subtract $3 x^{2}$ from $6 x^{2}$ .

$3 x^{2} - 5 x = 2$

Step 2

Subtract $2$ from both sides of the equation.

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

Step 3

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 2 = - 6$ and whose sum is $b = - 5$ .

Factor $- 5$ out of $- 5 x$ .

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

Rewrite $- 5$ as $1$ plus $- 6$

$3 x^{2} + (1 - 6) x - 2 = 0$

Apply the distributive property.

$3 x^{2} + 1 x - 6 x - 2 = 0$

Multiply $x$ by $1$ .

$3 x^{2} + x - 6 x - 2 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(3 x^{2} + x) - 6 x - 2 = 0$

Factor out the greatest common factor (GCF) from each group.

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

Factor the polynomial by factoring out the greatest common factor, $3 x + 1$ .

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

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$3 x + 1 = 0$

$x - 2 = 0$

Step 5

Set $3 x + 1$ equal to $0$ and solve for $x$ .

Set $3 x + 1$ equal to $0$ .

$3 x + 1 = 0$

Solve $3 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Divide each term in $3 x = - 1$ by $3$ and simplify.

Divide each term in $3 x = - 1$ by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 6

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Step 7

The final solution is all the values that make $(3 x + 1) (x - 2) = 0$ true.

$x = - \frac{1}{3}, 2$

Step 8

Find the Roots (Zeros) 6x^2-5x=3x^2+2

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