Solve for x 4x^2=14x+8

$4 x^{2} = 14 x + 8$

Step 1

Subtract $14 x$ from both sides of the equation.

$4 x^{2} - 14 x = 8$

Step 2

Subtract $8$ from both sides of the equation.

$4 x^{2} - 14 x - 8 = 0$

Step 3

Factor the left side of the equation.

Factor $2$ out of $4 x^{2} - 14 x - 8$ .

Factor $2$ out of $4 x^{2}$ .

$2 (2 x^{2}) - 14 x - 8 = 0$

Factor $2$ out of $- 14 x$ .

$2 (2 x^{2}) + 2 (- 7 x) - 8 = 0$

Factor $2$ out of $- 8$ .

$2 (2 x^{2}) + 2 (- 7 x) + 2 (- 4) = 0$

Factor $2$ out of $2 (2 x^{2}) + 2 (- 7 x)$ .

$2 (2 x^{2} - 7 x) + 2 (- 4) = 0$

Factor $2$ out of $2 (2 x^{2} - 7 x) + 2 (- 4)$ .

$2 (2 x^{2} - 7 x - 4) = 0$

Factor.

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 $a \cdot c = 2 \cdot - 4 = - 8$ and whose sum is $b = - 7$ .

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

$2 (2 x^{2} - 7 x - 4) = 0$

Rewrite $- 7$ as $1$ plus $- 8$

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

Apply the distributive property.

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

Multiply $x$ by $1$ .

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

$2 (x (2 x + 1) - 4 (2 x + 1)) = 0$

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

$2 ((2 x + 1) (x - 4)) = 0$

Remove unnecessary parentheses.

$2 (2 x + 1) (x - 4) = 0$

Step 4

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

$2 x + 1 = 0$

$x - 4 = 0$

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 $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

Step 7

The final solution is all the values that make $2 (2 x + 1) (x - 4) = 0$ true.

$x = - \frac{1}{2}, 4$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{2}, 4$

Decimal Form:

$x = - 0.5, 4$

Solve for x 4x^2=14x+8

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