Solve x=25/2*(17+x)^2/121

$$x = \frac { 25 } { 2 } \times \frac { ( 17 + x ) ^ { 2 } } { 121 }$$

Solve for x (complex solution)

$$x=-\frac{304}{25}+\frac{297}{25}i=-12.16+11.88i$$

$$x=-\frac{304}{25}-\frac{297}{25}i=-12.16-11.88i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(17+x\right)^{2}.

$$x=\frac{25}{2}\times \frac{289+34x+x^{2}}{121}$$

Multiply \frac{25}{2} times \frac{289+34x+x^{2}}{121} by multiplying numerator times numerator and denominator times denominator.

$$x=\frac{25\left(289+34x+x^{2}\right)}{2\times 121}$$

Multiply 2 and 121 to get 242.

$$x=\frac{25\left(289+34x+x^{2}\right)}{242}$$

Subtract \frac{25\left(289+34x+x^{2}\right)}{242} from both sides.

$$x-\frac{25\left(289+34x+x^{2}\right)}{242}=0$$

Use the distributive property to multiply 25 by 289+34x+x^{2}.

$$x-\frac{7225+850x+25x^{2}}{242}=0$$

Divide each term of 7225+850x+25x^{2} by 242 to get \frac{7225}{242}+\frac{425}{121}x+\frac{25}{242}x^{2}.

$$x-\left(\frac{7225}{242}+\frac{425}{121}x+\frac{25}{242}x^{2}\right)=0$$

To find the opposite of \frac{7225}{242}+\frac{425}{121}x+\frac{25}{242}x^{2}, find the opposite of each term.

$$x-\frac{7225}{242}-\frac{425}{121}x-\frac{25}{242}x^{2}=0$$

Combine x and -\frac{425}{121}x to get -\frac{304}{121}x.

$$-\frac{304}{121}x-\frac{7225}{242}-\frac{25}{242}x^{2}=0$$

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.

$$-\frac{25}{242}x^{2}-\frac{304}{121}x-\frac{7225}{242}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{25}{242} for a, -\frac{304}{121} for b, and -\frac{7225}{242} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{304}{121}\right)±\sqrt{\left(-\frac{304}{121}\right)^{2}-4\left(-\frac{25}{242}\right)\left(-\frac{7225}{242}\right)}}{2\left(-\frac{25}{242}\right)}$$

Square -\frac{304}{121} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{304}{121}\right)±\sqrt{\frac{92416}{14641}-4\left(-\frac{25}{242}\right)\left(-\frac{7225}{242}\right)}}{2\left(-\frac{25}{242}\right)}$$

Multiply -4 times -\frac{25}{242}.

$$x=\frac{-\left(-\frac{304}{121}\right)±\sqrt{\frac{92416}{14641}+\frac{50}{121}\left(-\frac{7225}{242}\right)}}{2\left(-\frac{25}{242}\right)}$$

Multiply \frac{50}{121} times -\frac{7225}{242} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{304}{121}\right)±\sqrt{\frac{92416-180625}{14641}}}{2\left(-\frac{25}{242}\right)}$$

Add \frac{92416}{14641} to -\frac{180625}{14641} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{304}{121}\right)±\sqrt{-\frac{729}{121}}}{2\left(-\frac{25}{242}\right)}$$

Take the square root of -\frac{729}{121}.

$$x=\frac{-\left(-\frac{304}{121}\right)±\frac{27}{11}i}{2\left(-\frac{25}{242}\right)}$$

The opposite of -\frac{304}{121} is \frac{304}{121}.

$$x=\frac{\frac{304}{121}±\frac{27}{11}i}{2\left(-\frac{25}{242}\right)}$$

Multiply 2 times -\frac{25}{242}.

$$x=\frac{\frac{304}{121}±\frac{27}{11}i}{-\frac{25}{121}}$$

Now solve the equation x=\frac{\frac{304}{121}±\frac{27}{11}i}{-\frac{25}{121}} when ± is plus. Add \frac{304}{121} to \frac{27}{11}i.

$$x=\frac{\frac{304}{121}+\frac{27}{11}i}{-\frac{25}{121}}$$

Divide \frac{304}{121}+\frac{27}{11}i by -\frac{25}{121} by multiplying \frac{304}{121}+\frac{27}{11}i by the reciprocal of -\frac{25}{121}.

$$x=-\frac{304}{25}-\frac{297}{25}i$$

Now solve the equation x=\frac{\frac{304}{121}±\frac{27}{11}i}{-\frac{25}{121}} when ± is minus. Subtract \frac{27}{11}i from \frac{304}{121}.

$$x=\frac{\frac{304}{121}-\frac{27}{11}i}{-\frac{25}{121}}$$

Divide \frac{304}{121}-\frac{27}{11}i by -\frac{25}{121} by multiplying \frac{304}{121}-\frac{27}{11}i by the reciprocal of -\frac{25}{121}.

$$x=-\frac{304}{25}+\frac{297}{25}i$$

The equation is now solved.

$$x=-\frac{304}{25}-\frac{297}{25}i x=-\frac{304}{25}+\frac{297}{25}i$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(17+x\right)^{2}.

$$x=\frac{25}{2}\times \frac{289+34x+x^{2}}{121}$$

Multiply \frac{25}{2} times \frac{289+34x+x^{2}}{121} by multiplying numerator times numerator and denominator times denominator.

$$x=\frac{25\left(289+34x+x^{2}\right)}{2\times 121}$$

Multiply 2 and 121 to get 242.

$$x=\frac{25\left(289+34x+x^{2}\right)}{242}$$

Subtract \frac{25\left(289+34x+x^{2}\right)}{242} from both sides.

$$x-\frac{25\left(289+34x+x^{2}\right)}{242}=0$$

Use the distributive property to multiply 25 by 289+34x+x^{2}.

$$x-\frac{7225+850x+25x^{2}}{242}=0$$

Divide each term of 7225+850x+25x^{2} by 242 to get \frac{7225}{242}+\frac{425}{121}x+\frac{25}{242}x^{2}.

$$x-\left(\frac{7225}{242}+\frac{425}{121}x+\frac{25}{242}x^{2}\right)=0$$

To find the opposite of \frac{7225}{242}+\frac{425}{121}x+\frac{25}{242}x^{2}, find the opposite of each term.

$$x-\frac{7225}{242}-\frac{425}{121}x-\frac{25}{242}x^{2}=0$$

Combine x and -\frac{425}{121}x to get -\frac{304}{121}x.

$$-\frac{304}{121}x-\frac{7225}{242}-\frac{25}{242}x^{2}=0$$

Add \frac{7225}{242} to both sides. Anything plus zero gives itself.

$$-\frac{304}{121}x-\frac{25}{242}x^{2}=\frac{7225}{242}$$

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.

$$-\frac{25}{242}x^{2}-\frac{304}{121}x=\frac{7225}{242}$$

Divide both sides of the equation by -\frac{25}{242}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{25}{242}x^{2}-\frac{304}{121}x}{-\frac{25}{242}}=\frac{\frac{7225}{242}}{-\frac{25}{242}}$$

Dividing by -\frac{25}{242} undoes the multiplication by -\frac{25}{242}.

$$x^{2}+\left(-\frac{\frac{304}{121}}{-\frac{25}{242}}\right)x=\frac{\frac{7225}{242}}{-\frac{25}{242}}$$

Divide -\frac{304}{121} by -\frac{25}{242} by multiplying -\frac{304}{121} by the reciprocal of -\frac{25}{242}.

$$x^{2}+\frac{608}{25}x=\frac{\frac{7225}{242}}{-\frac{25}{242}}$$

Divide \frac{7225}{242} by -\frac{25}{242} by multiplying \frac{7225}{242} by the reciprocal of -\frac{25}{242}.

$$x^{2}+\frac{608}{25}x=-289$$

Divide \frac{608}{25}, the coefficient of the x term, by 2 to get \frac{304}{25}. Then add the square of \frac{304}{25} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{608}{25}x+\left(\frac{304}{25}\right)^{2}=-289+\left(\frac{304}{25}\right)^{2}$$

Square \frac{304}{25} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{608}{25}x+\frac{92416}{625}=-289+\frac{92416}{625}$$

Add -289 to \frac{92416}{625}.

$$x^{2}+\frac{608}{25}x+\frac{92416}{625}=-\frac{88209}{625}$$

Factor x^{2}+\frac{608}{25}x+\frac{92416}{625}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{304}{25}\right)^{2}=-\frac{88209}{625}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{304}{25}\right)^{2}}=\sqrt{-\frac{88209}{625}}$$

Simplify.

$$x+\frac{304}{25}=\frac{297}{25}i x+\frac{304}{25}=-\frac{297}{25}i$$

Subtract \frac{304}{25} from both sides of the equation.

$$x=-\frac{304}{25}+\frac{297}{25}i x=-\frac{304}{25}-\frac{297}{25}i$$