Solve x(2/3+(x-1)4/3)/2=54

$$\frac{ x( \frac{ 2 }{ 3 } +(x-1) \frac{ 4 }{ 3 } ) }{ 2 } =54$$

Solve for x

$$x = \frac{\sqrt{1297} + 1}{4} \approx 9.253471553$$

$$x=\frac{1-\sqrt{1297}}{4}\approx -8.753471553$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$x\left(\frac{2}{3}+\left(x-1\right)\times \frac{4}{3}\right)=54\times 2$$

Use the distributive property to multiply x-1 by \frac{4}{3}.

$$x\left(\frac{2}{3}+x\times \frac{4}{3}-\frac{4}{3}\right)=54\times 2$$

Since \frac{2}{3} and \frac{4}{3} have the same denominator, subtract them by subtracting their numerators.

$$x\left(\frac{2-4}{3}+x\times \frac{4}{3}\right)=54\times 2$$

Subtract 4 from 2 to get -2.

$$x\left(-\frac{2}{3}+x\times \frac{4}{3}\right)=54\times 2$$

Use the distributive property to multiply x by -\frac{2}{3}+x\times \frac{4}{3}.

$$x\left(-\frac{2}{3}\right)+xx\times \frac{4}{3}=54\times 2$$

Multiply x and x to get x^{2}.

$$x\left(-\frac{2}{3}\right)+x^{2}\times \frac{4}{3}=54\times 2$$

Multiply 54 and 2 to get 108.

$$x\left(-\frac{2}{3}\right)+x^{2}\times \frac{4}{3}=108$$

Subtract 108 from both sides.

$$x\left(-\frac{2}{3}\right)+x^{2}\times \frac{4}{3}-108=0$$

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.

$$\frac{4}{3}x^{2}-\frac{2}{3}x-108=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{4}{3} for a, -\frac{2}{3} for b, and -108 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{2}{3}\right)±\sqrt{\left(-\frac{2}{3}\right)^{2}-4\times \frac{4}{3}\left(-108\right)}}{2\times \frac{4}{3}}$$

Square -\frac{2}{3} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{2}{3}\right)±\sqrt{\frac{4}{9}-4\times \frac{4}{3}\left(-108\right)}}{2\times \frac{4}{3}}$$

Multiply -4 times \frac{4}{3}.

$$x=\frac{-\left(-\frac{2}{3}\right)±\sqrt{\frac{4}{9}-\frac{16}{3}\left(-108\right)}}{2\times \frac{4}{3}}$$

Multiply -\frac{16}{3} times -108.

$$x=\frac{-\left(-\frac{2}{3}\right)±\sqrt{\frac{4}{9}+576}}{2\times \frac{4}{3}}$$

Add \frac{4}{9} to 576.

$$x=\frac{-\left(-\frac{2}{3}\right)±\sqrt{\frac{5188}{9}}}{2\times \frac{4}{3}}$$

Take the square root of \frac{5188}{9}.

$$x=\frac{-\left(-\frac{2}{3}\right)±\frac{2\sqrt{1297}}{3}}{2\times \frac{4}{3}}$$

The opposite of -\frac{2}{3} is \frac{2}{3}.

$$x=\frac{\frac{2}{3}±\frac{2\sqrt{1297}}{3}}{2\times \frac{4}{3}}$$

Multiply 2 times \frac{4}{3}.

$$x=\frac{\frac{2}{3}±\frac{2\sqrt{1297}}{3}}{\frac{8}{3}}$$

Now solve the equation x=\frac{\frac{2}{3}±\frac{2\sqrt{1297}}{3}}{\frac{8}{3}} when ± is plus. Add \frac{2}{3} to \frac{2\sqrt{1297}}{3}.

$$x=\frac{2\sqrt{1297}+2}{\frac{8}{3}\times 3}$$

Divide \frac{2+2\sqrt{1297}}{3} by \frac{8}{3} by multiplying \frac{2+2\sqrt{1297}}{3} by the reciprocal of \frac{8}{3}.

$$x=\frac{\sqrt{1297}+1}{4}$$

Now solve the equation x=\frac{\frac{2}{3}±\frac{2\sqrt{1297}}{3}}{\frac{8}{3}} when ± is minus. Subtract \frac{2\sqrt{1297}}{3} from \frac{2}{3}.

$$x=\frac{2-2\sqrt{1297}}{\frac{8}{3}\times 3}$$

Divide \frac{2-2\sqrt{1297}}{3} by \frac{8}{3} by multiplying \frac{2-2\sqrt{1297}}{3} by the reciprocal of \frac{8}{3}.

$$x=\frac{1-\sqrt{1297}}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{1297}+1}{4} x=\frac{1-\sqrt{1297}}{4}$$

Steps for Completing the Square

Multiply both sides by 2.

$$x\left(\frac{2}{3}+\left(x-1\right)\times \frac{4}{3}\right)=54\times 2$$

Use the distributive property to multiply x-1 by \frac{4}{3}.

$$x\left(\frac{2}{3}+x\times \frac{4}{3}-\frac{4}{3}\right)=54\times 2$$

Since \frac{2}{3} and \frac{4}{3} have the same denominator, subtract them by subtracting their numerators.

$$x\left(\frac{2-4}{3}+x\times \frac{4}{3}\right)=54\times 2$$

Subtract 4 from 2 to get -2.

$$x\left(-\frac{2}{3}+x\times \frac{4}{3}\right)=54\times 2$$

Use the distributive property to multiply x by -\frac{2}{3}+x\times \frac{4}{3}.

$$x\left(-\frac{2}{3}\right)+xx\times \frac{4}{3}=54\times 2$$

Multiply x and x to get x^{2}.

$$x\left(-\frac{2}{3}\right)+x^{2}\times \frac{4}{3}=54\times 2$$

Multiply 54 and 2 to get 108.

$$x\left(-\frac{2}{3}\right)+x^{2}\times \frac{4}{3}=108$$

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.

$$\frac{4}{3}x^{2}-\frac{2}{3}x=108$$

Divide both sides of the equation by \frac{4}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{4}{3}x^{2}-\frac{2}{3}x}{\frac{4}{3}}=\frac{108}{\frac{4}{3}}$$

Dividing by \frac{4}{3} undoes the multiplication by \frac{4}{3}.

$$x^{2}+\left(-\frac{\frac{2}{3}}{\frac{4}{3}}\right)x=\frac{108}{\frac{4}{3}}$$

Divide -\frac{2}{3} by \frac{4}{3} by multiplying -\frac{2}{3} by the reciprocal of \frac{4}{3}.

$$x^{2}-\frac{1}{2}x=\frac{108}{\frac{4}{3}}$$

Divide 108 by \frac{4}{3} by multiplying 108 by the reciprocal of \frac{4}{3}.

$$x^{2}-\frac{1}{2}x=81$$

Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=81+\left(-\frac{1}{4}\right)^{2}$$

Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=81+\frac{1}{16}$$

Add 81 to \frac{1}{16}.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{1297}{16}$$

Factor x^{2}-\frac{1}{2}x+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{1297}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{1297}{16}}$$

Simplify.

$$x-\frac{1}{4}=\frac{\sqrt{1297}}{4} x-\frac{1}{4}=-\frac{\sqrt{1297}}{4}$$

Add \frac{1}{4} to both sides of the equation.

$$x=\frac{\sqrt{1297}+1}{4} x=\frac{1-\sqrt{1297}}{4}$$