Solve 62/2+1=221/2left(x+2right)+221/2left(x-2right)

$$\frac{ 62 }{ 2 } +1 = \frac{ 221 }{ 2 \left( x+2 \right) } + \frac{ 221 }{ 2 \left( x-2 \right) }$$

Solve for x

$$x = \frac{5 \sqrt{2609} + 221}{64} \approx 7.443622747$$

$$x=\frac{221-5\sqrt{2609}}{64}\approx -0.537372747$$

Steps Using the Quadratic Formula

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

$$\left(x^{2}-4\right)\times 62+2\left(x-2\right)\left(x+2\right)=\left(x-2\right)\times 221+\left(x+2\right)\times 221$$

Use the distributive property to multiply x^{2}-4 by 62.

$$62x^{2}-248+2\left(x-2\right)\left(x+2\right)=\left(x-2\right)\times 221+\left(x+2\right)\times 221$$

Use the distributive property to multiply 2 by x-2.

$$62x^{2}-248+\left(2x-4\right)\left(x+2\right)=\left(x-2\right)\times 221+\left(x+2\right)\times 221$$

Use the distributive property to multiply 2x-4 by x+2 and combine like terms.

$$62x^{2}-248+2x^{2}-8=\left(x-2\right)\times 221+\left(x+2\right)\times 221$$

Combine 62x^{2} and 2x^{2} to get 64x^{2}.

$$64x^{2}-248-8=\left(x-2\right)\times 221+\left(x+2\right)\times 221$$

Subtract 8 from -248 to get -256.

$$64x^{2}-256=\left(x-2\right)\times 221+\left(x+2\right)\times 221$$

Use the distributive property to multiply x-2 by 221.

$$64x^{2}-256=221x-442+\left(x+2\right)\times 221$$

Use the distributive property to multiply x+2 by 221.

$$64x^{2}-256=221x-442+221x+442$$

Combine 221x and 221x to get 442x.

$$64x^{2}-256=442x-442+442$$

Add -442 and 442 to get 0.

$$64x^{2}-256=442x$$

Subtract 442x from both sides.

$$64x^{2}-256-442x=0$$

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.

$$64x^{2}-442x-256=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 64 for a, -442 for b, and -256 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-442\right)±\sqrt{\left(-442\right)^{2}-4\times 64\left(-256\right)}}{2\times 64}$$

Square -442.

$$x=\frac{-\left(-442\right)±\sqrt{195364-4\times 64\left(-256\right)}}{2\times 64}$$

Multiply -4 times 64.

$$x=\frac{-\left(-442\right)±\sqrt{195364-256\left(-256\right)}}{2\times 64}$$

Multiply -256 times -256.

$$x=\frac{-\left(-442\right)±\sqrt{195364+65536}}{2\times 64}$$

Add 195364 to 65536.

$$x=\frac{-\left(-442\right)±\sqrt{260900}}{2\times 64}$$

Take the square root of 260900.

$$x=\frac{-\left(-442\right)±10\sqrt{2609}}{2\times 64}$$

The opposite of -442 is 442.

$$x=\frac{442±10\sqrt{2609}}{2\times 64}$$

Multiply 2 times 64.

$$x=\frac{442±10\sqrt{2609}}{128}$$

Now solve the equation x=\frac{442±10\sqrt{2609}}{128} when ± is plus. Add 442 to 10\sqrt{2609}.

$$x=\frac{10\sqrt{2609}+442}{128}$$

Divide 442+10\sqrt{2609} by 128.

$$x=\frac{5\sqrt{2609}+221}{64}$$

Now solve the equation x=\frac{442±10\sqrt{2609}}{128} when ± is minus. Subtract 10\sqrt{2609} from 442.

$$x=\frac{442-10\sqrt{2609}}{128}$$

Divide 442-10\sqrt{2609} by 128.

$$x=\frac{221-5\sqrt{2609}}{64}$$

The equation is now solved.

$$x=\frac{5\sqrt{2609}+221}{64} x=\frac{221-5\sqrt{2609}}{64}$$

Steps for Completing the Square

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

$$\left(x^{2}-4\right)\times 62+2\left(x-2\right)\left(x+2\right)=\left(x-2\right)\times 221+\left(x+2\right)\times 221$$

Use the distributive property to multiply x^{2}-4 by 62.

$$62x^{2}-248+2\left(x-2\right)\left(x+2\right)=\left(x-2\right)\times 221+\left(x+2\right)\times 221$$

Use the distributive property to multiply 2 by x-2.

$$62x^{2}-248+\left(2x-4\right)\left(x+2\right)=\left(x-2\right)\times 221+\left(x+2\right)\times 221$$

Use the distributive property to multiply 2x-4 by x+2 and combine like terms.

$$62x^{2}-248+2x^{2}-8=\left(x-2\right)\times 221+\left(x+2\right)\times 221$$

Combine 62x^{2} and 2x^{2} to get 64x^{2}.

$$64x^{2}-248-8=\left(x-2\right)\times 221+\left(x+2\right)\times 221$$

Subtract 8 from -248 to get -256.

$$64x^{2}-256=\left(x-2\right)\times 221+\left(x+2\right)\times 221$$

Use the distributive property to multiply x-2 by 221.

$$64x^{2}-256=221x-442+\left(x+2\right)\times 221$$

Use the distributive property to multiply x+2 by 221.

$$64x^{2}-256=221x-442+221x+442$$

Combine 221x and 221x to get 442x.

$$64x^{2}-256=442x-442+442$$

Add -442 and 442 to get 0.

$$64x^{2}-256=442x$$

Subtract 442x from both sides.

$$64x^{2}-256-442x=0$$

Add 256 to both sides. Anything plus zero gives itself.

$$64x^{2}-442x=256$$

Divide both sides by 64.

$$\frac{64x^{2}-442x}{64}=\frac{256}{64}$$

Dividing by 64 undoes the multiplication by 64.

$$x^{2}+\left(-\frac{442}{64}\right)x=\frac{256}{64}$$

Reduce the fraction \frac{-442}{64} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{221}{32}x=\frac{256}{64}$$

Divide 256 by 64.

$$x^{2}-\frac{221}{32}x=4$$

Divide -\frac{221}{32}, the coefficient of the x term, by 2 to get -\frac{221}{64}. Then add the square of -\frac{221}{64} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{221}{32}x+\left(-\frac{221}{64}\right)^{2}=4+\left(-\frac{221}{64}\right)^{2}$$

Square -\frac{221}{64} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{221}{32}x+\frac{48841}{4096}=4+\frac{48841}{4096}$$

Add 4 to \frac{48841}{4096}.

$$x^{2}-\frac{221}{32}x+\frac{48841}{4096}=\frac{65225}{4096}$$

Factor x^{2}-\frac{221}{32}x+\frac{48841}{4096}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{221}{64}\right)^{2}=\frac{65225}{4096}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{221}{64}\right)^{2}}=\sqrt{\frac{65225}{4096}}$$

Simplify.

$$x-\frac{221}{64}=\frac{5\sqrt{2609}}{64} x-\frac{221}{64}=-\frac{5\sqrt{2609}}{64}$$

Add \frac{221}{64} to both sides of the equation.

$$x=\frac{5\sqrt{2609}+221}{64} x=\frac{221-5\sqrt{2609}}{64}$$