Solve x/x^2-4x-1=7

$$\frac { x } { x ^ { 2 } - 4 x - 1 } = 7$$

Solve for x

$$x = \frac{\sqrt{1037} + 29}{14} \approx 4.371606027$$

$$x=\frac{29-\sqrt{1037}}{14}\approx -0.228748884$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by \left(x-\left(\sqrt{5}+2\right)\right)\left(x-\left(-\sqrt{5}+2\right)\right).

$$x=7\left(x-\left(\sqrt{5}+2\right)\right)\left(x-\left(-\sqrt{5}+2\right)\right)$$

To find the opposite of \sqrt{5}+2, find the opposite of each term.

$$x=7\left(x-\sqrt{5}-2\right)\left(x-\left(-\sqrt{5}+2\right)\right)$$

To find the opposite of -\sqrt{5}+2, find the opposite of each term.

$$x=7\left(x-\sqrt{5}-2\right)\left(x+\sqrt{5}-2\right)$$

Use the distributive property to multiply 7 by x-\sqrt{5}-2.

$$x=\left(7x-7\sqrt{5}-14\right)\left(x+\sqrt{5}-2\right)$$

Use the distributive property to multiply 7x-7\sqrt{5}-14 by x+\sqrt{5}-2 and combine like terms.

$$x=7x^{2}-28x-7\left(\sqrt{5}\right)^{2}+28$$

The square of \sqrt{5} is 5.

$$x=7x^{2}-28x-7\times 5+28$$

Multiply -7 and 5 to get -35.

$$x=7x^{2}-28x-35+28$$

Add -35 and 28 to get -7.

$$x=7x^{2}-28x-7$$

Subtract 7x^{2} from both sides.

$$x-7x^{2}=-28x-7$$

Add 28x to both sides.

$$x-7x^{2}+28x=-7$$

Combine x and 28x to get 29x.

$$29x-7x^{2}=-7$$

Add 7 to both sides.

$$29x-7x^{2}+7=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-7x^{2}+29x+7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 29 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-29±\sqrt{29^{2}-4\left(-7\right)\times 7}}{2\left(-7\right)}$$

Square 29.

$$x=\frac{-29±\sqrt{841-4\left(-7\right)\times 7}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-29±\sqrt{841+28\times 7}}{2\left(-7\right)}$$

Multiply 28 times 7.

$$x=\frac{-29±\sqrt{841+196}}{2\left(-7\right)}$$

Add 841 to 196.

$$x=\frac{-29±\sqrt{1037}}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{-29±\sqrt{1037}}{-14}$$

Now solve the equation x=\frac{-29±\sqrt{1037}}{-14} when ± is plus. Add -29 to \sqrt{1037}.

$$x=\frac{\sqrt{1037}-29}{-14}$$

Divide -29+\sqrt{1037} by -14.

$$x=\frac{29-\sqrt{1037}}{14}$$

Now solve the equation x=\frac{-29±\sqrt{1037}}{-14} when ± is minus. Subtract \sqrt{1037} from -29.

$$x=\frac{-\sqrt{1037}-29}{-14}$$

Divide -29-\sqrt{1037} by -14.

$$x=\frac{\sqrt{1037}+29}{14}$$

The equation is now solved.

$$x=\frac{29-\sqrt{1037}}{14} x=\frac{\sqrt{1037}+29}{14}$$

Steps for Completing the Square

Multiply both sides of the equation by \left(x-\left(\sqrt{5}+2\right)\right)\left(x-\left(-\sqrt{5}+2\right)\right).

$$x=7\left(x-\left(\sqrt{5}+2\right)\right)\left(x-\left(-\sqrt{5}+2\right)\right)$$

To find the opposite of \sqrt{5}+2, find the opposite of each term.

$$x=7\left(x-\sqrt{5}-2\right)\left(x-\left(-\sqrt{5}+2\right)\right)$$

To find the opposite of -\sqrt{5}+2, find the opposite of each term.

$$x=7\left(x-\sqrt{5}-2\right)\left(x+\sqrt{5}-2\right)$$

Use the distributive property to multiply 7 by x-\sqrt{5}-2.

$$x=\left(7x-7\sqrt{5}-14\right)\left(x+\sqrt{5}-2\right)$$

Use the distributive property to multiply 7x-7\sqrt{5}-14 by x+\sqrt{5}-2 and combine like terms.

$$x=7x^{2}-28x-7\left(\sqrt{5}\right)^{2}+28$$

The square of \sqrt{5} is 5.

$$x=7x^{2}-28x-7\times 5+28$$

Multiply -7 and 5 to get -35.

$$x=7x^{2}-28x-35+28$$

Add -35 and 28 to get -7.

$$x=7x^{2}-28x-7$$

Subtract 7x^{2} from both sides.

$$x-7x^{2}=-28x-7$$

Add 28x to both sides.

$$x-7x^{2}+28x=-7$$

Combine x and 28x to get 29x.

$$29x-7x^{2}=-7$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-7x^{2}+29x=-7$$

Divide both sides by -7.

$$\frac{-7x^{2}+29x}{-7}=-\frac{7}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$x^{2}+\frac{29}{-7}x=-\frac{7}{-7}$$

Divide 29 by -7.

$$x^{2}-\frac{29}{7}x=-\frac{7}{-7}$$

Divide -7 by -7.

$$x^{2}-\frac{29}{7}x=1$$

Divide -\frac{29}{7}, the coefficient of the x term, by 2 to get -\frac{29}{14}. Then add the square of -\frac{29}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{29}{7}x+\left(-\frac{29}{14}\right)^{2}=1+\left(-\frac{29}{14}\right)^{2}$$

Square -\frac{29}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{29}{7}x+\frac{841}{196}=1+\frac{841}{196}$$

Add 1 to \frac{841}{196}.

$$x^{2}-\frac{29}{7}x+\frac{841}{196}=\frac{1037}{196}$$

Factor x^{2}-\frac{29}{7}x+\frac{841}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{29}{14}\right)^{2}=\frac{1037}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{29}{14}\right)^{2}}=\sqrt{\frac{1037}{196}}$$

Simplify.

$$x-\frac{29}{14}=\frac{\sqrt{1037}}{14} x-\frac{29}{14}=-\frac{\sqrt{1037}}{14}$$

Add \frac{29}{14} to both sides of the equation.

$$x=\frac{\sqrt{1037}+29}{14} x=\frac{29-\sqrt{1037}}{14}$$