Solve 8x[3+(2-3+x/5)+1]-3=x

$$8 x [ 3 + ( 2 - \frac { 3 + x } { 5 } ) + 1 ] - 3 = x$$

Solve for x

$$x = \frac{\sqrt{44041} + 211}{16} \approx 26.3037173$$

$$x=\frac{211-\sqrt{44041}}{16}\approx 0.0712827$$

Steps Using the Quadratic Formula

Add 3 and 2 to get 5.

$$8x\left(5-\frac{3+x}{5}+1\right)-3=x$$

Add 5 and 1 to get 6.

$$8x\left(6-\frac{3+x}{5}\right)-3=x$$

Use the distributive property to multiply 8x by 6-\frac{3+x}{5}.

$$48x+8x\left(-\frac{3+x}{5}\right)-3=x$$

Express 8\left(-\frac{3+x}{5}\right) as a single fraction.

$$48x+\frac{-8\left(3+x\right)}{5}x-3=x$$

Express \frac{-8\left(3+x\right)}{5}x as a single fraction.

$$48x+\frac{-8\left(3+x\right)x}{5}-3=x$$

Use the distributive property to multiply -8 by 3+x.

$$48x+\frac{\left(-24-8x\right)x}{5}-3=x$$

Use the distributive property to multiply -24-8x by x.

$$48x+\frac{-24x-8x^{2}}{5}-3=x$$

Divide each term of -24x-8x^{2} by 5 to get -\frac{24}{5}x-\frac{8}{5}x^{2}.

$$48x-\frac{24}{5}x-\frac{8}{5}x^{2}-3=x$$

Combine 48x and -\frac{24}{5}x to get \frac{216}{5}x.

$$\frac{216}{5}x-\frac{8}{5}x^{2}-3=x$$

Subtract x from both sides.

$$\frac{216}{5}x-\frac{8}{5}x^{2}-3-x=0$$

Combine \frac{216}{5}x and -x to get \frac{211}{5}x.

$$\frac{211}{5}x-\frac{8}{5}x^{2}-3=0$$

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.

$$-\frac{8}{5}x^{2}+\frac{211}{5}x-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{8}{5} for a, \frac{211}{5} for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{211}{5}±\sqrt{\left(\frac{211}{5}\right)^{2}-4\left(-\frac{8}{5}\right)\left(-3\right)}}{2\left(-\frac{8}{5}\right)}$$

Square \frac{211}{5} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{211}{5}±\sqrt{\frac{44521}{25}-4\left(-\frac{8}{5}\right)\left(-3\right)}}{2\left(-\frac{8}{5}\right)}$$

Multiply -4 times -\frac{8}{5}.

$$x=\frac{-\frac{211}{5}±\sqrt{\frac{44521}{25}+\frac{32}{5}\left(-3\right)}}{2\left(-\frac{8}{5}\right)}$$

Multiply \frac{32}{5} times -3.

$$x=\frac{-\frac{211}{5}±\sqrt{\frac{44521}{25}-\frac{96}{5}}}{2\left(-\frac{8}{5}\right)}$$

Add \frac{44521}{25} to -\frac{96}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{211}{5}±\sqrt{\frac{44041}{25}}}{2\left(-\frac{8}{5}\right)}$$

Take the square root of \frac{44041}{25}.

$$x=\frac{-\frac{211}{5}±\frac{\sqrt{44041}}{5}}{2\left(-\frac{8}{5}\right)}$$

Multiply 2 times -\frac{8}{5}.

$$x=\frac{-\frac{211}{5}±\frac{\sqrt{44041}}{5}}{-\frac{16}{5}}$$

Now solve the equation x=\frac{-\frac{211}{5}±\frac{\sqrt{44041}}{5}}{-\frac{16}{5}} when ± is plus. Add -\frac{211}{5} to \frac{\sqrt{44041}}{5}.

$$x=\frac{\sqrt{44041}-211}{-\frac{16}{5}\times 5}$$

Divide \frac{-211+\sqrt{44041}}{5} by -\frac{16}{5} by multiplying \frac{-211+\sqrt{44041}}{5} by the reciprocal of -\frac{16}{5}.

$$x=\frac{211-\sqrt{44041}}{16}$$

Now solve the equation x=\frac{-\frac{211}{5}±\frac{\sqrt{44041}}{5}}{-\frac{16}{5}} when ± is minus. Subtract \frac{\sqrt{44041}}{5} from -\frac{211}{5}.

$$x=\frac{-\sqrt{44041}-211}{-\frac{16}{5}\times 5}$$

Divide \frac{-211-\sqrt{44041}}{5} by -\frac{16}{5} by multiplying \frac{-211-\sqrt{44041}}{5} by the reciprocal of -\frac{16}{5}.

$$x=\frac{\sqrt{44041}+211}{16}$$

The equation is now solved.

$$x=\frac{211-\sqrt{44041}}{16} x=\frac{\sqrt{44041}+211}{16}$$

Steps for Completing the Square

Add 3 and 2 to get 5.

$$8x\left(5-\frac{3+x}{5}+1\right)-3=x$$

Add 5 and 1 to get 6.

$$8x\left(6-\frac{3+x}{5}\right)-3=x$$

Use the distributive property to multiply 8x by 6-\frac{3+x}{5}.

$$48x+8x\left(-\frac{3+x}{5}\right)-3=x$$

Express 8\left(-\frac{3+x}{5}\right) as a single fraction.

$$48x+\frac{-8\left(3+x\right)}{5}x-3=x$$

Express \frac{-8\left(3+x\right)}{5}x as a single fraction.

$$48x+\frac{-8\left(3+x\right)x}{5}-3=x$$

Use the distributive property to multiply -8 by 3+x.

$$48x+\frac{\left(-24-8x\right)x}{5}-3=x$$

Use the distributive property to multiply -24-8x by x.

$$48x+\frac{-24x-8x^{2}}{5}-3=x$$

Divide each term of -24x-8x^{2} by 5 to get -\frac{24}{5}x-\frac{8}{5}x^{2}.

$$48x-\frac{24}{5}x-\frac{8}{5}x^{2}-3=x$$

Combine 48x and -\frac{24}{5}x to get \frac{216}{5}x.

$$\frac{216}{5}x-\frac{8}{5}x^{2}-3=x$$

Subtract x from both sides.

$$\frac{216}{5}x-\frac{8}{5}x^{2}-3-x=0$$

Combine \frac{216}{5}x and -x to get \frac{211}{5}x.

$$\frac{211}{5}x-\frac{8}{5}x^{2}-3=0$$

Add 3 to both sides. Anything plus zero gives itself.

$$\frac{211}{5}x-\frac{8}{5}x^{2}=3$$

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.

$$-\frac{8}{5}x^{2}+\frac{211}{5}x=3$$

Divide both sides of the equation by -\frac{8}{5}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{8}{5}x^{2}+\frac{211}{5}x}{-\frac{8}{5}}=\frac{3}{-\frac{8}{5}}$$

Dividing by -\frac{8}{5} undoes the multiplication by -\frac{8}{5}.

$$x^{2}+\frac{\frac{211}{5}}{-\frac{8}{5}}x=\frac{3}{-\frac{8}{5}}$$

Divide \frac{211}{5} by -\frac{8}{5} by multiplying \frac{211}{5} by the reciprocal of -\frac{8}{5}.

$$x^{2}-\frac{211}{8}x=\frac{3}{-\frac{8}{5}}$$

Divide 3 by -\frac{8}{5} by multiplying 3 by the reciprocal of -\frac{8}{5}.

$$x^{2}-\frac{211}{8}x=-\frac{15}{8}$$

Divide -\frac{211}{8}, the coefficient of the x term, by 2 to get -\frac{211}{16}. Then add the square of -\frac{211}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{211}{8}x+\left(-\frac{211}{16}\right)^{2}=-\frac{15}{8}+\left(-\frac{211}{16}\right)^{2}$$

Square -\frac{211}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{211}{8}x+\frac{44521}{256}=-\frac{15}{8}+\frac{44521}{256}$$

Add -\frac{15}{8} to \frac{44521}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{211}{8}x+\frac{44521}{256}=\frac{44041}{256}$$

Factor x^{2}-\frac{211}{8}x+\frac{44521}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{211}{16}\right)^{2}=\frac{44041}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{211}{16}\right)^{2}}=\sqrt{\frac{44041}{256}}$$

Simplify.

$$x-\frac{211}{16}=\frac{\sqrt{44041}}{16} x-\frac{211}{16}=-\frac{\sqrt{44041}}{16}$$

Add \frac{211}{16} to both sides of the equation.

$$x=\frac{\sqrt{44041}+211}{16} x=\frac{211-\sqrt{44041}}{16}$$