Solve (a_{1}+d)^2=a_{1}(a_{1}+3a)

$$( a _ { 1 } + d ) ^ { 2 } = a _ { 1 } ( a _ { 1 } + 3 a )$$

Solve for a

$$\left\{\begin{matrix}a=\frac{d\left(2a_{1}+d\right)}{3a_{1}}\text{, }&a_{1}\neq 0\\a\in \mathrm{R}\text{, }&a_{1}=0\text{ and }d=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(a_{1}+d\right)^{2}.

$$a_{1}^{2}+2a_{1}d+d^{2}=a_{1}\left(a_{1}+3a\right)$$

Use the distributive property to multiply a_{1} by a_{1}+3a.

$$a_{1}^{2}+2a_{1}d+d^{2}=a_{1}^{2}+3a_{1}a$$

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

$$a_{1}^{2}+3a_{1}a=a_{1}^{2}+2a_{1}d+d^{2}$$

Subtract a_{1}^{2} from both sides.

$$3a_{1}a=a_{1}^{2}+2a_{1}d+d^{2}-a_{1}^{2}$$

Combine a_{1}^{2} and -a_{1}^{2} to get 0.

$$3a_{1}a=2a_{1}d+d^{2}$$

Divide both sides by 3a_{1}.

$$\frac{3a_{1}a}{3a_{1}}=\frac{d\left(2a_{1}+d\right)}{3a_{1}}$$

Dividing by 3a_{1} undoes the multiplication by 3a_{1}.

$$a=\frac{d\left(2a_{1}+d\right)}{3a_{1}}$$

Solve for a_1

$$\left\{\begin{matrix}a_{1}=-\frac{d^{2}}{2d-3a}\text{, }&d\neq \frac{3a}{2}\\a_{1}\in \mathrm{R}\text{, }&d=0\text{ and }a=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(a_{1}+d\right)^{2}.

$$a_{1}^{2}+2a_{1}d+d^{2}=a_{1}\left(a_{1}+3a\right)$$

Use the distributive property to multiply a_{1} by a_{1}+3a.

$$a_{1}^{2}+2a_{1}d+d^{2}=a_{1}^{2}+3a_{1}a$$

Subtract a_{1}^{2} from both sides.

$$a_{1}^{2}+2a_{1}d+d^{2}-a_{1}^{2}=3a_{1}a$$

Combine a_{1}^{2} and -a_{1}^{2} to get 0.

$$2a_{1}d+d^{2}=3a_{1}a$$

Subtract 3a_{1}a from both sides.

$$2a_{1}d+d^{2}-3a_{1}a=0$$

Subtract d^{2} from both sides. Anything subtracted from zero gives its negation.

$$2a_{1}d-3a_{1}a=-d^{2}$$

Combine all terms containing a_{1}.

$$\left(2d-3a\right)a_{1}=-d^{2}$$

Divide both sides by 2d-3a.

$$\frac{\left(2d-3a\right)a_{1}}{2d-3a}=-\frac{d^{2}}{2d-3a}$$

Dividing by 2d-3a undoes the multiplication by 2d-3a.

$$a_{1}=-\frac{d^{2}}{2d-3a}$$