Solve (3.1+25/6x)^2+x^2=24.8^2

$$( 3.1 + \frac { 25 } { 6 } x ) ^ { 2 } + x ^ { 2 } = 24.8 ^ { 2 }$$

Solve for x

$$x=\frac{186\sqrt{10567}}{3305}-\frac{465}{661}\approx 5.081706524$$

$$x=-\frac{186\sqrt{10567}}{3305}-\frac{465}{661}\approx -6.488665677$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3.1+\frac{25}{6}x\right)^{2}.

$$9.61+\frac{155}{6}x+\frac{625}{36}x^{2}+x^{2}=24.8^{2}$$

Combine \frac{625}{36}x^{2} and x^{2} to get \frac{661}{36}x^{2}.

$$9.61+\frac{155}{6}x+\frac{661}{36}x^{2}=24.8^{2}$$

Calculate 24.8 to the power of 2 and get 615.04.

$$9.61+\frac{155}{6}x+\frac{661}{36}x^{2}=615.04$$

Subtract 615.04 from both sides.

$$9.61+\frac{155}{6}x+\frac{661}{36}x^{2}-615.04=0$$

Subtract 615.04 from 9.61 to get -605.43.

$$-605.43+\frac{155}{6}x+\frac{661}{36}x^{2}=0$$

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.

$$\frac{661}{36}x^{2}+\frac{155}{6}x-605.43=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{661}{36} for a, \frac{155}{6} for b, and -605.43 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{155}{6}±\sqrt{\left(\frac{155}{6}\right)^{2}-4\times \frac{661}{36}\left(-605.43\right)}}{2\times \frac{661}{36}}$$

Square \frac{155}{6} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{155}{6}±\sqrt{\frac{24025}{36}-4\times \frac{661}{36}\left(-605.43\right)}}{2\times \frac{661}{36}}$$

Multiply -4 times \frac{661}{36}.

$$x=\frac{-\frac{155}{6}±\sqrt{\frac{24025}{36}-\frac{661}{9}\left(-605.43\right)}}{2\times \frac{661}{36}}$$

Multiply -\frac{661}{9} times -605.43 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{155}{6}±\sqrt{\frac{24025}{36}+\frac{4446547}{100}}}{2\times \frac{661}{36}}$$

Add \frac{24025}{36} to \frac{4446547}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{155}{6}±\sqrt{\frac{10154887}{225}}}{2\times \frac{661}{36}}$$

Take the square root of \frac{10154887}{225}.

$$x=\frac{-\frac{155}{6}±\frac{31\sqrt{10567}}{15}}{2\times \frac{661}{36}}$$

Multiply 2 times \frac{661}{36}.

$$x=\frac{-\frac{155}{6}±\frac{31\sqrt{10567}}{15}}{\frac{661}{18}}$$

Now solve the equation x=\frac{-\frac{155}{6}±\frac{31\sqrt{10567}}{15}}{\frac{661}{18}} when ± is plus. Add -\frac{155}{6} to \frac{31\sqrt{10567}}{15}.

$$x=\frac{\frac{31\sqrt{10567}}{15}-\frac{155}{6}}{\frac{661}{18}}$$

Divide -\frac{155}{6}+\frac{31\sqrt{10567}}{15} by \frac{661}{18} by multiplying -\frac{155}{6}+\frac{31\sqrt{10567}}{15} by the reciprocal of \frac{661}{18}.

$$x=\frac{186\sqrt{10567}}{3305}-\frac{465}{661}$$

Now solve the equation x=\frac{-\frac{155}{6}±\frac{31\sqrt{10567}}{15}}{\frac{661}{18}} when ± is minus. Subtract \frac{31\sqrt{10567}}{15} from -\frac{155}{6}.

$$x=\frac{-\frac{31\sqrt{10567}}{15}-\frac{155}{6}}{\frac{661}{18}}$$

Divide -\frac{155}{6}-\frac{31\sqrt{10567}}{15} by \frac{661}{18} by multiplying -\frac{155}{6}-\frac{31\sqrt{10567}}{15} by the reciprocal of \frac{661}{18}.

$$x=-\frac{186\sqrt{10567}}{3305}-\frac{465}{661}$$

The equation is now solved.

$$x=\frac{186\sqrt{10567}}{3305}-\frac{465}{661} x=-\frac{186\sqrt{10567}}{3305}-\frac{465}{661}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3.1+\frac{25}{6}x\right)^{2}.

$$9.61+\frac{155}{6}x+\frac{625}{36}x^{2}+x^{2}=24.8^{2}$$

Combine \frac{625}{36}x^{2} and x^{2} to get \frac{661}{36}x^{2}.

$$9.61+\frac{155}{6}x+\frac{661}{36}x^{2}=24.8^{2}$$

Calculate 24.8 to the power of 2 and get 615.04.

$$9.61+\frac{155}{6}x+\frac{661}{36}x^{2}=615.04$$

Subtract 9.61 from both sides.

$$\frac{155}{6}x+\frac{661}{36}x^{2}=615.04-9.61$$

Subtract 9.61 from 615.04 to get 605.43.

$$\frac{155}{6}x+\frac{661}{36}x^{2}=605.43$$

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.

$$\frac{661}{36}x^{2}+\frac{155}{6}x=605.43$$

Divide both sides of the equation by \frac{661}{36}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{661}{36}x^{2}+\frac{155}{6}x}{\frac{661}{36}}=\frac{605.43}{\frac{661}{36}}$$

Dividing by \frac{661}{36} undoes the multiplication by \frac{661}{36}.

$$x^{2}+\frac{\frac{155}{6}}{\frac{661}{36}}x=\frac{605.43}{\frac{661}{36}}$$

Divide \frac{155}{6} by \frac{661}{36} by multiplying \frac{155}{6} by the reciprocal of \frac{661}{36}.

$$x^{2}+\frac{930}{661}x=\frac{605.43}{\frac{661}{36}}$$

Divide 605.43 by \frac{661}{36} by multiplying 605.43 by the reciprocal of \frac{661}{36}.

$$x^{2}+\frac{930}{661}x=\frac{544887}{16525}$$

Divide \frac{930}{661}, the coefficient of the x term, by 2 to get \frac{465}{661}. Then add the square of \frac{465}{661} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{930}{661}x+\left(\frac{465}{661}\right)^{2}=\frac{544887}{16525}+\left(\frac{465}{661}\right)^{2}$$

Square \frac{465}{661} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{930}{661}x+\frac{216225}{436921}=\frac{544887}{16525}+\frac{216225}{436921}$$

Add \frac{544887}{16525} to \frac{216225}{436921} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{930}{661}x+\frac{216225}{436921}=\frac{365575932}{10923025}$$

Factor x^{2}+\frac{930}{661}x+\frac{216225}{436921}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{465}{661}\right)^{2}=\frac{365575932}{10923025}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{465}{661}\right)^{2}}=\sqrt{\frac{365575932}{10923025}}$$

Simplify.

$$x+\frac{465}{661}=\frac{186\sqrt{10567}}{3305} x+\frac{465}{661}=-\frac{186\sqrt{10567}}{3305}$$

Subtract \frac{465}{661} from both sides of the equation.

$$x=\frac{186\sqrt{10567}}{3305}-\frac{465}{661} x=-\frac{186\sqrt{10567}}{3305}-\frac{465}{661}$$