Solve by Factoring 2x^2-8=31x-2x^2

$2 x^{2} - 8 = 31 x - 2 x^{2}$

Step 1

Move all the expressions to the left side of the equation.

Subtract $31 x$ from both sides of the equation.

$2 x^{2} - 8 - 31 x = - 2 x^{2}$

Add $2 x^{2}$ to both sides of the equation.

$2 x^{2} - 8 - 31 x + 2 x^{2} = 0$

Step 2

Add $2 x^{2}$ and $2 x^{2}$ .

$4 x^{2} - 8 - 31 x = 0$

Step 3

Factor by grouping.

Reorder terms.

$4 x^{2} - 31 x - 8 = 0$

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 $a \cdot c = 4 \cdot - 8 = - 32$ and whose sum is $b = - 31$ .

Factor $- 31$ out of $- 31 x$ .

$4 x^{2} - 31 x - 8 = 0$

Rewrite $- 31$ as $1$ plus $- 32$

$4 x^{2} + (1 - 32) x - 8 = 0$

Apply the distributive property.

$4 x^{2} + 1 x - 32 x - 8 = 0$

Multiply $x$ by $1$ .

$4 x^{2} + x - 32 x - 8 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(4 x^{2} + x) - 32 x - 8 = 0$

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

$x (4 x + 1) - 8 (4 x + 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $4 x + 1$ .

$(4 x + 1) (x - 8) = 0$

Step 4

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

$4 x + 1 = 0$

$x - 8 = 0$

Step 5

Set $4 x + 1$ equal to $0$ and solve for $x$ .

Set $4 x + 1$ equal to $0$ .

$4 x + 1 = 0$

Solve $4 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$4 x = - 1$

Divide each term in $4 x = - 1$ by $4$ and simplify.

Divide each term in $4 x = - 1$ by $4$ .

$\frac{4 x}{4} = \frac{- 1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 1}{4}$

Divide $x$ by $1$ .

$x = \frac{- 1}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{4}$

Step 6

Set $x - 8$ equal to $0$ and solve for $x$ .

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

Step 7

The final solution is all the values that make $(4 x + 1) (x - 8) = 0$ true.

$x = - \frac{1}{4}, 8$

Solve by Factoring 2x^2-8=31x-2x^2

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