Solve 7x-113/6x+42=5x+116/x+256

$$\frac { 7 x - 113 } { 6 x + 42 } = \frac { 5 x + 116 } { x + 256 }$$

Solve for x (complex solution)

$$x=\frac{-3\sqrt{279119}i+773}{46}\approx 16.804347826-34.455465626i$$

$$x=\frac{773+3\sqrt{279119}i}{46}\approx 16.804347826+34.455465626i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -256,-7 since division by zero is not defined. Multiply both sides of the equation by 6\left(x+7\right)\left(x+256\right), the least common multiple of 6x+42,x+256.

$$\left(x+256\right)\left(7x-113\right)=\left(6x+42\right)\left(5x+116\right)$$

Use the distributive property to multiply x+256 by 7x-113 and combine like terms.

$$7x^{2}+1679x-28928=\left(6x+42\right)\left(5x+116\right)$$

Use the distributive property to multiply 6x+42 by 5x+116 and combine like terms.

$$7x^{2}+1679x-28928=30x^{2}+906x+4872$$

Subtract 30x^{2} from both sides.

$$7x^{2}+1679x-28928-30x^{2}=906x+4872$$

Combine 7x^{2} and -30x^{2} to get -23x^{2}.

$$-23x^{2}+1679x-28928=906x+4872$$

Subtract 906x from both sides.

$$-23x^{2}+1679x-28928-906x=4872$$

Combine 1679x and -906x to get 773x.

$$-23x^{2}+773x-28928=4872$$

Subtract 4872 from both sides.

$$-23x^{2}+773x-28928-4872=0$$

Subtract 4872 from -28928 to get -33800.

$$-23x^{2}+773x-33800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -23 for a, 773 for b, and -33800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-773±\sqrt{773^{2}-4\left(-23\right)\left(-33800\right)}}{2\left(-23\right)}$$

Square 773.

$$x=\frac{-773±\sqrt{597529-4\left(-23\right)\left(-33800\right)}}{2\left(-23\right)}$$

Multiply -4 times -23.

$$x=\frac{-773±\sqrt{597529+92\left(-33800\right)}}{2\left(-23\right)}$$

Multiply 92 times -33800.

$$x=\frac{-773±\sqrt{597529-3109600}}{2\left(-23\right)}$$

Add 597529 to -3109600.

$$x=\frac{-773±\sqrt{-2512071}}{2\left(-23\right)}$$

Take the square root of -2512071.

$$x=\frac{-773±3\sqrt{279119}i}{2\left(-23\right)}$$

Multiply 2 times -23.

$$x=\frac{-773±3\sqrt{279119}i}{-46}$$

Now solve the equation x=\frac{-773±3\sqrt{279119}i}{-46} when ± is plus. Add -773 to 3i\sqrt{279119}.

$$x=\frac{-773+3\sqrt{279119}i}{-46}$$

Divide -773+3i\sqrt{279119} by -46.

$$x=\frac{-3\sqrt{279119}i+773}{46}$$

Now solve the equation x=\frac{-773±3\sqrt{279119}i}{-46} when ± is minus. Subtract 3i\sqrt{279119} from -773.

$$x=\frac{-3\sqrt{279119}i-773}{-46}$$

Divide -773-3i\sqrt{279119} by -46.

$$x=\frac{773+3\sqrt{279119}i}{46}$$

The equation is now solved.

$$x=\frac{-3\sqrt{279119}i+773}{46} x=\frac{773+3\sqrt{279119}i}{46}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -256,-7 since division by zero is not defined. Multiply both sides of the equation by 6\left(x+7\right)\left(x+256\right), the least common multiple of 6x+42,x+256.

$$\left(x+256\right)\left(7x-113\right)=\left(6x+42\right)\left(5x+116\right)$$

Use the distributive property to multiply x+256 by 7x-113 and combine like terms.

$$7x^{2}+1679x-28928=\left(6x+42\right)\left(5x+116\right)$$

Use the distributive property to multiply 6x+42 by 5x+116 and combine like terms.

$$7x^{2}+1679x-28928=30x^{2}+906x+4872$$

Subtract 30x^{2} from both sides.

$$7x^{2}+1679x-28928-30x^{2}=906x+4872$$

Combine 7x^{2} and -30x^{2} to get -23x^{2}.

$$-23x^{2}+1679x-28928=906x+4872$$

Subtract 906x from both sides.

$$-23x^{2}+1679x-28928-906x=4872$$

Combine 1679x and -906x to get 773x.

$$-23x^{2}+773x-28928=4872$$

Add 28928 to both sides.

$$-23x^{2}+773x=4872+28928$$

Add 4872 and 28928 to get 33800.

$$-23x^{2}+773x=33800$$

Divide both sides by -23.

$$\frac{-23x^{2}+773x}{-23}=\frac{33800}{-23}$$

Dividing by -23 undoes the multiplication by -23.

$$x^{2}+\frac{773}{-23}x=\frac{33800}{-23}$$

Divide 773 by -23.

$$x^{2}-\frac{773}{23}x=\frac{33800}{-23}$$

Divide 33800 by -23.

$$x^{2}-\frac{773}{23}x=-\frac{33800}{23}$$

Divide -\frac{773}{23}, the coefficient of the x term, by 2 to get -\frac{773}{46}. Then add the square of -\frac{773}{46} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{773}{23}x+\left(-\frac{773}{46}\right)^{2}=-\frac{33800}{23}+\left(-\frac{773}{46}\right)^{2}$$

Square -\frac{773}{46} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{773}{23}x+\frac{597529}{2116}=-\frac{33800}{23}+\frac{597529}{2116}$$

Add -\frac{33800}{23} to \frac{597529}{2116} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{773}{23}x+\frac{597529}{2116}=-\frac{2512071}{2116}$$

Factor x^{2}-\frac{773}{23}x+\frac{597529}{2116}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{773}{46}\right)^{2}=-\frac{2512071}{2116}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{773}{46}\right)^{2}}=\sqrt{-\frac{2512071}{2116}}$$

Simplify.

$$x-\frac{773}{46}=\frac{3\sqrt{279119}i}{46} x-\frac{773}{46}=-\frac{3\sqrt{279119}i}{46}$$

Add \frac{773}{46} to both sides of the equation.

$$x=\frac{773+3\sqrt{279119}i}{46} x=\frac{-3\sqrt{279119}i+773}{46}$$