Find the Roots (Zeros) 6x^2-7x+2=7

$6 x^{2} - 7 x + 2 = 7$

Step 1

Subtract $7$ from both sides of the equation.

$6 x^{2} - 7 x + 2 - 7 = 0$

Step 2

Subtract $7$ from $2$ .

$6 x^{2} - 7 x - 5 = 0$

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 $a \cdot c = 6 \cdot - 5 = - 30$ and whose sum is $b = - 7$ .

Factor $- 7$ out of $- 7 x$ .

$6 x^{2} - 7 x - 5 = 0$

Rewrite $- 7$ as $3$ plus $- 10$

$6 x^{2} + (3 - 10) x - 5 = 0$

Apply the distributive property.

$6 x^{2} + 3 x - 10 x - 5 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(6 x^{2} + 3 x) - 10 x - 5 = 0$

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

$3 x (2 x + 1) - 5 (2 x + 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$(2 x + 1) (3 x - 5) = 0$

Step 4

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

$2 x + 1 = 0$

$3 x - 5 = 0$

Step 5

Set $2 x + 1$ equal to $0$ and solve for $x$ .

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Solve $2 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Divide each term in $2 x = - 1$ by $2$ and simplify.

Divide each term in $2 x = - 1$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 6

Set $3 x - 5$ equal to $0$ and solve for $x$ .

Set $3 x - 5$ equal to $0$ .

$3 x - 5 = 0$

Solve $3 x - 5 = 0$ for $x$ .

Add $5$ to both sides of the equation.

$3 x = 5$

Divide each term in $3 x = 5$ by $3$ and simplify.

Divide each term in $3 x = 5$ by $3$ .

$\frac{3 x}{3} = \frac{5}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{5}{3}$

Divide $x$ by $1$ .

$x = \frac{5}{3}$

Step 7

The final solution is all the values that make $(2 x + 1) (3 x - 5) = 0$ true.

$x = - \frac{1}{2}, \frac{5}{3}$

Step 8

Find the Roots (Zeros) 6x^2-7x+2=7

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