Solve {left(2x+5/3xright)}^2=k+7/2k

$${ \left( \frac{ 2x+5 }{ 3x } \right) }^{ 2 } = \frac{ k+7 }{ 2k }$$

Solve for k

$$k=\frac{63x^{2}}{50+40x-x^{2}}$$

$$x\neq 0\text{ and }x\neq 15\sqrt{2}+20\text{ and }x\neq 20-15\sqrt{2}$$

Steps for Solving Linear Equation

Variable k cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2k.

$$2k\times \left(\frac{2x+5}{3x}\right)^{2}=k+7$$

To raise \frac{2x+5}{3x} to a power, raise both numerator and denominator to the power and then divide.

$$2k\times \frac{\left(2x+5\right)^{2}}{\left(3x\right)^{2}}=k+7$$

Express 2\times \frac{\left(2x+5\right)^{2}}{\left(3x\right)^{2}} as a single fraction.

$$\frac{2\left(2x+5\right)^{2}}{\left(3x\right)^{2}}k=k+7$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+5\right)^{2}.

$$\frac{2\left(4x^{2}+20x+25\right)}{\left(3x\right)^{2}}k=k+7$$

Expand \left(3x\right)^{2}.

$$\frac{2\left(4x^{2}+20x+25\right)}{3^{2}x^{2}}k=k+7$$

Calculate 3 to the power of 2 and get 9.

$$\frac{2\left(4x^{2}+20x+25\right)}{9x^{2}}k=k+7$$

Express \frac{2\left(4x^{2}+20x+25\right)}{9x^{2}}k as a single fraction.

$$\frac{2\left(4x^{2}+20x+25\right)k}{9x^{2}}=k+7$$

Subtract k from both sides.

$$\frac{2\left(4x^{2}+20x+25\right)k}{9x^{2}}-k=7$$

Use the distributive property to multiply 2 by 4x^{2}+20x+25.

$$\frac{\left(8x^{2}+40x+50\right)k}{9x^{2}}-k=7$$

Use the distributive property to multiply 8x^{2}+40x+50 by k.

$$\frac{8x^{2}k+40xk+50k}{9x^{2}}-k=7$$

To add or subtract expressions, expand them to make their denominators the same. Multiply k times \frac{9x^{2}}{9x^{2}}.

$$\frac{8x^{2}k+40xk+50k}{9x^{2}}-\frac{k\times 9x^{2}}{9x^{2}}=7$$

Since \frac{8x^{2}k+40xk+50k}{9x^{2}} and \frac{k\times 9x^{2}}{9x^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{8x^{2}k+40xk+50k-k\times 9x^{2}}{9x^{2}}=7$$

Do the multiplications in 8x^{2}k+40xk+50k-k\times 9x^{2}.

$$\frac{8x^{2}k+40xk+50k-9x^{2}k}{9x^{2}}=7$$

Combine like terms in 8x^{2}k+40xk+50k-9x^{2}k.

$$\frac{50k-x^{2}k+40xk}{9x^{2}}=7$$

Multiply both sides of the equation by 9x^{2}.

$$50k-x^{2}k+40xk=63x^{2}$$

Combine all terms containing k.

$$\left(50-x^{2}+40x\right)k=63x^{2}$$

The equation is in standard form.

$$\left(50+40x-x^{2}\right)k=63x^{2}$$

Divide both sides by -x^{2}+40x+50.

$$\frac{\left(50+40x-x^{2}\right)k}{50+40x-x^{2}}=\frac{63x^{2}}{50+40x-x^{2}}$$

Dividing by -x^{2}+40x+50 undoes the multiplication by -x^{2}+40x+50.

$$k=\frac{63x^{2}}{50+40x-x^{2}}$$

Variable k cannot be equal to 0.

$$k=\frac{63x^{2}}{50+40x-x^{2}}\text{, }k\neq 0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{5\left(3\sqrt{\frac{2\left(k+7\right)}{k}}+4\right)k}{k+63}\text{, }&k\neq -63\text{ and }k\neq 0\\x=-\frac{5\sqrt{2}}{3\sqrt{\frac{k+7}{k}}+2\sqrt{2}}\text{, }&k\neq 0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{10}{3\sqrt{\frac{2\left(k+7\right)}{k}}+4}\text{, }&k\leq -7\\x=\frac{5\left(3\sqrt{\frac{2\left(k+7\right)}{k}}+4\right)k}{k+63}\text{, }&k\leq -7\text{ and }k\neq -63\\x=\frac{5\sqrt{k}\left(3\sqrt{2\left(k+7\right)}+4\sqrt{k}\right)}{k+63}\text{; }x=-\frac{10\sqrt{k}}{3\sqrt{2\left(k+7\right)}+4\sqrt{k}}\text{, }&k>0\end{matrix}\right.$$