Solve (25+100d)^2=2(x*(21+1)ad^2)

$$( 25 + 100 d ) ^ { 2 } = 2 ( x \cdot ( 21 + 1 ) a d ^ { 2 } )$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{625\left(4+\frac{1}{d}\right)^{2}}{44x}\text{, }&d\neq 0\text{ and }x\neq 0\\a\in \mathrm{C}\text{, }&d=-\frac{1}{4}\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(25+100d\right)^{2}.

$$625+5000d+10000d^{2}=2x\left(21+1\right)ad^{2}$$

Add 21 and 1 to get 22.

$$625+5000d+10000d^{2}=2x\times 22ad^{2}$$

Multiply 2 and 22 to get 44.

$$625+5000d+10000d^{2}=44xad^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$44xad^{2}=625+5000d+10000d^{2}$$

The equation is in standard form.

$$44xd^{2}a=10000d^{2}+5000d+625$$

Divide both sides by 44xd^{2}.

$$\frac{44xd^{2}a}{44xd^{2}}=\frac{625\left(4d+1\right)^{2}}{44xd^{2}}$$

Dividing by 44xd^{2} undoes the multiplication by 44xd^{2}.

$$a=\frac{625\left(4d+1\right)^{2}}{44xd^{2}}$$

Solve for a

$$\left\{\begin{matrix}a=\frac{625\left(4+\frac{1}{d}\right)^{2}}{44x}\text{, }&d\neq 0\text{ and }x\neq 0\\a\in \mathrm{R}\text{, }&d=-\frac{1}{4}\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(25+100d\right)^{2}.

$$625+5000d+10000d^{2}=2x\left(21+1\right)ad^{2}$$

Add 21 and 1 to get 22.

$$625+5000d+10000d^{2}=2x\times 22ad^{2}$$

Multiply 2 and 22 to get 44.

$$625+5000d+10000d^{2}=44xad^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$44xad^{2}=625+5000d+10000d^{2}$$

The equation is in standard form.

$$44xd^{2}a=10000d^{2}+5000d+625$$

Divide both sides by 44xd^{2}.

$$\frac{44xd^{2}a}{44xd^{2}}=\frac{625\left(4d+1\right)^{2}}{44xd^{2}}$$

Dividing by 44xd^{2} undoes the multiplication by 44xd^{2}.

$$a=\frac{625\left(4d+1\right)^{2}}{44xd^{2}}$$

Solve for d (complex solution)

$$\left\{\begin{matrix}d=\frac{25\left(\sqrt{11ax}-50\right)}{2\left(2500-11ax\right)}\text{; }d=-\frac{25\left(\sqrt{11ax}+50\right)}{2\left(2500-11ax\right)}\text{, }&a=0\text{ or }x\neq \frac{2500}{11a}\\d=-\frac{1}{8}\text{, }&x=\frac{2500}{11a}\text{ and }a\neq 0\end{matrix}\right.$$

Solve for d

$$\left\{\begin{matrix}d=\frac{25\left(\sqrt{11ax}-50\right)}{2\left(2500-11ax\right)}\text{; }d=-\frac{25\left(\sqrt{11ax}+50\right)}{2\left(2500-11ax\right)}\text{, }&\left(x\neq \frac{2500}{11a}\text{ and }x\leq 0\text{ and }a\leq 0\right)\text{ or }\left(x\neq \frac{2500}{11a}\text{ and }a\geq 0\text{ and }x\geq 0\right)\text{ or }a=0\\d=-\frac{1}{8}\text{, }&x=\frac{2500}{11a}\text{ and }a\neq 0\end{matrix}\right.$$