Solve x^2-14x+41/x^2-49=3/17

$$\frac { x ^ { 2 } - 14 x + 41 } { x ^ { 2 } - 49 } = \frac { 3 } { 17 }$$

Solve for x

$$x=\frac{\sqrt{2345}}{14}+\frac{17}{2}\approx 11.958942861$$

$$x=-\frac{\sqrt{2345}}{14}+\frac{17}{2}\approx 5.041057139$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -7,7 since division by zero is not defined. Multiply both sides of the equation by 17\left(x-7\right)\left(x+7\right), the least common multiple of x^{2}-49,17.

$$17\left(x^{2}-14x+41\right)=3\left(x-7\right)\left(x+7\right)$$

Use the distributive property to multiply 17 by x^{2}-14x+41.

$$17x^{2}-238x+697=3\left(x-7\right)\left(x+7\right)$$

Use the distributive property to multiply 3 by x-7.

$$17x^{2}-238x+697=\left(3x-21\right)\left(x+7\right)$$

Use the distributive property to multiply 3x-21 by x+7 and combine like terms.

$$17x^{2}-238x+697=3x^{2}-147$$

Subtract 3x^{2} from both sides.

$$17x^{2}-238x+697-3x^{2}=-147$$

Combine 17x^{2} and -3x^{2} to get 14x^{2}.

$$14x^{2}-238x+697=-147$$

Add 147 to both sides.

$$14x^{2}-238x+697+147=0$$

Add 697 and 147 to get 844.

$$14x^{2}-238x+844=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 14 for a, -238 for b, and 844 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-238\right)±\sqrt{\left(-238\right)^{2}-4\times 14\times 844}}{2\times 14}$$

Square -238.

$$x=\frac{-\left(-238\right)±\sqrt{56644-4\times 14\times 844}}{2\times 14}$$

Multiply -4 times 14.

$$x=\frac{-\left(-238\right)±\sqrt{56644-56\times 844}}{2\times 14}$$

Multiply -56 times 844.

$$x=\frac{-\left(-238\right)±\sqrt{56644-47264}}{2\times 14}$$

Add 56644 to -47264.

$$x=\frac{-\left(-238\right)±\sqrt{9380}}{2\times 14}$$

Take the square root of 9380.

$$x=\frac{-\left(-238\right)±2\sqrt{2345}}{2\times 14}$$

The opposite of -238 is 238.

$$x=\frac{238±2\sqrt{2345}}{2\times 14}$$

Multiply 2 times 14.

$$x=\frac{238±2\sqrt{2345}}{28}$$

Now solve the equation x=\frac{238±2\sqrt{2345}}{28} when ± is plus. Add 238 to 2\sqrt{2345}.

$$x=\frac{2\sqrt{2345}+238}{28}$$

Divide 238+2\sqrt{2345} by 28.

$$x=\frac{\sqrt{2345}}{14}+\frac{17}{2}$$

Now solve the equation x=\frac{238±2\sqrt{2345}}{28} when ± is minus. Subtract 2\sqrt{2345} from 238.

$$x=\frac{238-2\sqrt{2345}}{28}$$

Divide 238-2\sqrt{2345} by 28.

$$x=-\frac{\sqrt{2345}}{14}+\frac{17}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{2345}}{14}+\frac{17}{2} x=-\frac{\sqrt{2345}}{14}+\frac{17}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -7,7 since division by zero is not defined. Multiply both sides of the equation by 17\left(x-7\right)\left(x+7\right), the least common multiple of x^{2}-49,17.

$$17\left(x^{2}-14x+41\right)=3\left(x-7\right)\left(x+7\right)$$

Use the distributive property to multiply 17 by x^{2}-14x+41.

$$17x^{2}-238x+697=3\left(x-7\right)\left(x+7\right)$$

Use the distributive property to multiply 3 by x-7.

$$17x^{2}-238x+697=\left(3x-21\right)\left(x+7\right)$$

Use the distributive property to multiply 3x-21 by x+7 and combine like terms.

$$17x^{2}-238x+697=3x^{2}-147$$

Subtract 3x^{2} from both sides.

$$17x^{2}-238x+697-3x^{2}=-147$$

Combine 17x^{2} and -3x^{2} to get 14x^{2}.

$$14x^{2}-238x+697=-147$$

Subtract 697 from both sides.

$$14x^{2}-238x=-147-697$$

Subtract 697 from -147 to get -844.

$$14x^{2}-238x=-844$$

Divide both sides by 14.

$$\frac{14x^{2}-238x}{14}=-\frac{844}{14}$$

Dividing by 14 undoes the multiplication by 14.

$$x^{2}+\left(-\frac{238}{14}\right)x=-\frac{844}{14}$$

Divide -238 by 14.

$$x^{2}-17x=-\frac{844}{14}$$

Reduce the fraction \frac{-844}{14} to lowest terms by extracting and canceling out 2.

$$x^{2}-17x=-\frac{422}{7}$$

Divide -17, the coefficient of the x term, by 2 to get -\frac{17}{2}. Then add the square of -\frac{17}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-17x+\left(-\frac{17}{2}\right)^{2}=-\frac{422}{7}+\left(-\frac{17}{2}\right)^{2}$$

Square -\frac{17}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-17x+\frac{289}{4}=-\frac{422}{7}+\frac{289}{4}$$

Add -\frac{422}{7} to \frac{289}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-17x+\frac{289}{4}=\frac{335}{28}$$

Factor x^{2}-17x+\frac{289}{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}.

$$\left(x-\frac{17}{2}\right)^{2}=\frac{335}{28}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{17}{2}\right)^{2}}=\sqrt{\frac{335}{28}}$$

Simplify.

$$x-\frac{17}{2}=\frac{\sqrt{2345}}{14} x-\frac{17}{2}=-\frac{\sqrt{2345}}{14}$$

Add \frac{17}{2} to both sides of the equation.

$$x=\frac{\sqrt{2345}}{14}+\frac{17}{2} x=-\frac{\sqrt{2345}}{14}+\frac{17}{2}$$