Solve for x 1=7x^2+6x

$1 = 7 x^{2} + 6 x$

Step 1

Rewrite the equation as $7 x^{2} + 6 x = 1$ .

$7 x^{2} + 6 x = 1$

Step 2

Subtract $1$ from both sides of the equation.

$7 x^{2} + 6 x - 1 = 0$

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 $a \cdot c = 7 \cdot - 1 = - 7$ and whose sum is $b = 6$ .

Factor $6$ out of $6 x$ .

$7 x^{2} + 6 (x) - 1 = 0$

Rewrite $6$ as $- 1$ plus $7$

$7 x^{2} + (- 1 + 7) x - 1 = 0$

Apply the distributive property.

$7 x^{2} - 1 x + 7 x - 1 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(7 x^{2} - 1 x) + 7 x - 1 = 0$

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

$x (7 x - 1) + 1 (7 x - 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $7 x - 1$ .

$(7 x - 1) (x + 1) = 0$

Step 4

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

$7 x - 1 = 0$

$x + 1 = 0$

Step 5

Set $7 x - 1$ equal to $0$ and solve for $x$ .

Set $7 x - 1$ equal to $0$ .

$7 x - 1 = 0$

Solve $7 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$7 x = 1$

Divide each term in $7 x = 1$ by $7$ and simplify.

Divide each term in $7 x = 1$ by $7$ .

$\frac{7 x}{7} = \frac{1}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x}{7} = \frac{1}{7}$

Divide $x$ by $1$ .

$x = \frac{1}{7}$

Step 6

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

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

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 7

The final solution is all the values that make $(7 x - 1) (x + 1) = 0$ true.

$x = \frac{1}{7}, - 1$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{7}, - 1$

Decimal Form:

$x = 0.\overline{142857}, - 1$

Solve for x 1=7x^2+6x

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