Solve x+(-1)=162/42+7x

$$x + ( - 1 ) = \frac { 162 } { 42 + 7 x }$$

Solve for x

$$x=\frac{\sqrt{6937}}{14}-\frac{5}{2}\approx 3.449189621$$

$$x=-\frac{\sqrt{6937}}{14}-\frac{5}{2}\approx -8.449189621$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -6 since division by zero is not defined. Multiply both sides of the equation by 7\left(x+6\right).

$$7\left(x+6\right)x+7\left(x+6\right)\left(-1\right)=162$$

Use the distributive property to multiply 7 by x+6.

$$\left(7x+42\right)x+7\left(x+6\right)\left(-1\right)=162$$

Use the distributive property to multiply 7x+42 by x.

$$7x^{2}+42x+7\left(x+6\right)\left(-1\right)=162$$

Multiply 7 and -1 to get -7.

$$7x^{2}+42x-7\left(x+6\right)=162$$

Use the distributive property to multiply -7 by x+6.

$$7x^{2}+42x-7x-42=162$$

Combine 42x and -7x to get 35x.

$$7x^{2}+35x-42=162$$

Subtract 162 from both sides.

$$7x^{2}+35x-42-162=0$$

Subtract 162 from -42 to get -204.

$$7x^{2}+35x-204=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, 35 for b, and -204 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-35±\sqrt{35^{2}-4\times 7\left(-204\right)}}{2\times 7}$$

Square 35.

$$x=\frac{-35±\sqrt{1225-4\times 7\left(-204\right)}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-35±\sqrt{1225-28\left(-204\right)}}{2\times 7}$$

Multiply -28 times -204.

$$x=\frac{-35±\sqrt{1225+5712}}{2\times 7}$$

Add 1225 to 5712.

$$x=\frac{-35±\sqrt{6937}}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{-35±\sqrt{6937}}{14}$$

Now solve the equation x=\frac{-35±\sqrt{6937}}{14} when ± is plus. Add -35 to \sqrt{6937}.

$$x=\frac{\sqrt{6937}-35}{14}$$

Divide -35+\sqrt{6937} by 14.

$$x=\frac{\sqrt{6937}}{14}-\frac{5}{2}$$

Now solve the equation x=\frac{-35±\sqrt{6937}}{14} when ± is minus. Subtract \sqrt{6937} from -35.

$$x=\frac{-\sqrt{6937}-35}{14}$$

Divide -35-\sqrt{6937} by 14.

$$x=-\frac{\sqrt{6937}}{14}-\frac{5}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{6937}}{14}-\frac{5}{2} x=-\frac{\sqrt{6937}}{14}-\frac{5}{2}$$

Steps for Completing the Square

Variable x cannot be equal to -6 since division by zero is not defined. Multiply both sides of the equation by 7\left(x+6\right).

$$7\left(x+6\right)x+7\left(x+6\right)\left(-1\right)=162$$

Use the distributive property to multiply 7 by x+6.

$$\left(7x+42\right)x+7\left(x+6\right)\left(-1\right)=162$$

Use the distributive property to multiply 7x+42 by x.

$$7x^{2}+42x+7\left(x+6\right)\left(-1\right)=162$$

Multiply 7 and -1 to get -7.

$$7x^{2}+42x-7\left(x+6\right)=162$$

Use the distributive property to multiply -7 by x+6.

$$7x^{2}+42x-7x-42=162$$

Combine 42x and -7x to get 35x.

$$7x^{2}+35x-42=162$$

Add 42 to both sides.

$$7x^{2}+35x=162+42$$

Add 162 and 42 to get 204.

$$7x^{2}+35x=204$$

Divide both sides by 7.

$$\frac{7x^{2}+35x}{7}=\frac{204}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$x^{2}+\frac{35}{7}x=\frac{204}{7}$$

Divide 35 by 7.

$$x^{2}+5x=\frac{204}{7}$$

Divide 5, the coefficient of the x term, by 2 to get \frac{5}{2}. Then add the square of \frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=\frac{204}{7}+\left(\frac{5}{2}\right)^{2}$$

Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+5x+\frac{25}{4}=\frac{204}{7}+\frac{25}{4}$$

Add \frac{204}{7} to \frac{25}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+5x+\frac{25}{4}=\frac{991}{28}$$

Factor x^{2}+5x+\frac{25}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{2}\right)^{2}=\frac{991}{28}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{\frac{991}{28}}$$

Simplify.

$$x+\frac{5}{2}=\frac{\sqrt{6937}}{14} x+\frac{5}{2}=-\frac{\sqrt{6937}}{14}$$

Subtract \frac{5}{2} from both sides of the equation.

$$x=\frac{\sqrt{6937}}{14}-\frac{5}{2} x=-\frac{\sqrt{6937}}{14}-\frac{5}{2}$$