Solve x^2-1/17x+7x+x^2=1/7

$$\frac { x ^ { 2 } - 1 } { 17 x + 7 x + x ^ { 2 } } = \frac { 1 } { 7 }$$

Solve for x

$$x=\frac{\sqrt{186}}{6}+2\approx 4.273030283$$

$$x=-\frac{\sqrt{186}}{6}+2\approx -0.273030283$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -24,0 since division by zero is not defined. Multiply both sides of the equation by 7x\left(x+24\right), the least common multiple of 17x+7x+x^{2},7.

$$7\left(x^{2}-1\right)=x\left(x+24\right)$$

Use the distributive property to multiply 7 by x^{2}-1.

$$7x^{2}-7=x\left(x+24\right)$$

Use the distributive property to multiply x by x+24.

$$7x^{2}-7=x^{2}+24x$$

Subtract x^{2} from both sides.

$$7x^{2}-7-x^{2}=24x$$

Combine 7x^{2} and -x^{2} to get 6x^{2}.

$$6x^{2}-7=24x$$

Subtract 24x from both sides.

$$6x^{2}-7-24x=0$$

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.

$$6x^{2}-24x-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -24 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 6\left(-7\right)}}{2\times 6}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 6\left(-7\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-24\right)±\sqrt{576-24\left(-7\right)}}{2\times 6}$$

Multiply -24 times -7.

$$x=\frac{-\left(-24\right)±\sqrt{576+168}}{2\times 6}$$

Add 576 to 168.

$$x=\frac{-\left(-24\right)±\sqrt{744}}{2\times 6}$$

Take the square root of 744.

$$x=\frac{-\left(-24\right)±2\sqrt{186}}{2\times 6}$$

The opposite of -24 is 24.

$$x=\frac{24±2\sqrt{186}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{24±2\sqrt{186}}{12}$$

Now solve the equation x=\frac{24±2\sqrt{186}}{12} when ± is plus. Add 24 to 2\sqrt{186}.

$$x=\frac{2\sqrt{186}+24}{12}$$

Divide 24+2\sqrt{186} by 12.

$$x=\frac{\sqrt{186}}{6}+2$$

Now solve the equation x=\frac{24±2\sqrt{186}}{12} when ± is minus. Subtract 2\sqrt{186} from 24.

$$x=\frac{24-2\sqrt{186}}{12}$$

Divide 24-2\sqrt{186} by 12.

$$x=-\frac{\sqrt{186}}{6}+2$$

The equation is now solved.

$$x=\frac{\sqrt{186}}{6}+2 x=-\frac{\sqrt{186}}{6}+2$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -24,0 since division by zero is not defined. Multiply both sides of the equation by 7x\left(x+24\right), the least common multiple of 17x+7x+x^{2},7.

$$7\left(x^{2}-1\right)=x\left(x+24\right)$$

Use the distributive property to multiply 7 by x^{2}-1.

$$7x^{2}-7=x\left(x+24\right)$$

Use the distributive property to multiply x by x+24.

$$7x^{2}-7=x^{2}+24x$$

Subtract x^{2} from both sides.

$$7x^{2}-7-x^{2}=24x$$

Combine 7x^{2} and -x^{2} to get 6x^{2}.

$$6x^{2}-7=24x$$

Subtract 24x from both sides.

$$6x^{2}-7-24x=0$$

Add 7 to both sides. Anything plus zero gives itself.

$$6x^{2}-24x=7$$

Divide both sides by 6.

$$\frac{6x^{2}-24x}{6}=\frac{7}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\left(-\frac{24}{6}\right)x=\frac{7}{6}$$

Divide -24 by 6.

$$x^{2}-4x=\frac{7}{6}$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-4x+\left(-2\right)^{2}=\frac{7}{6}+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=\frac{7}{6}+4$$

Add \frac{7}{6} to 4.

$$x^{2}-4x+4=\frac{31}{6}$$

Factor x^{2}-4x+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}.

$$\left(x-2\right)^{2}=\frac{31}{6}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{\frac{31}{6}}$$

Simplify.

$$x-2=\frac{\sqrt{186}}{6} x-2=-\frac{\sqrt{186}}{6}$$

Add 2 to both sides of the equation.

$$x=\frac{\sqrt{186}}{6}+2 x=-\frac{\sqrt{186}}{6}+2$$