Solve (a+3d)^2=a(a+12d)

$$( a + 3 d ) ^ { 2 } = a ( a + 12 d )$$

Solve for a (complex solution)

$$\left\{\begin{matrix}\\a=\frac{3d}{2}\text{, }&\text{unconditionally}\\a\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(a+3d\right)^{2}.

$$a^{2}+6ad+9d^{2}=a\left(a+12d\right)$$

Use the distributive property to multiply a by a+12d.

$$a^{2}+6ad+9d^{2}=a^{2}+12ad$$

Subtract a^{2} from both sides.

$$a^{2}+6ad+9d^{2}-a^{2}=12ad$$

Combine a^{2} and -a^{2} to get 0.

$$6ad+9d^{2}=12ad$$

Subtract 12ad from both sides.

$$6ad+9d^{2}-12ad=0$$

Combine 6ad and -12ad to get -6ad.

$$-6ad+9d^{2}=0$$

Subtract 9d^{2} from both sides. Anything subtracted from zero gives its negation.

$$-6ad=-9d^{2}$$

The equation is in standard form.

$$\left(-6d\right)a=-9d^{2}$$

Divide both sides by -6d.

$$\frac{\left(-6d\right)a}{-6d}=-\frac{9d^{2}}{-6d}$$

Dividing by -6d undoes the multiplication by -6d.

$$a=-\frac{9d^{2}}{-6d}$$

Divide -9d^{2} by -6d.

$$a=\frac{3d}{2}$$

Solve for a

$$\left\{\begin{matrix}\\a=\frac{3d}{2}\text{, }&\text{unconditionally}\\a\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(a+3d\right)^{2}.

$$a^{2}+6ad+9d^{2}=a\left(a+12d\right)$$

Use the distributive property to multiply a by a+12d.

$$a^{2}+6ad+9d^{2}=a^{2}+12ad$$

Subtract a^{2} from both sides.

$$a^{2}+6ad+9d^{2}-a^{2}=12ad$$

Combine a^{2} and -a^{2} to get 0.

$$6ad+9d^{2}=12ad$$

Subtract 12ad from both sides.

$$6ad+9d^{2}-12ad=0$$

Combine 6ad and -12ad to get -6ad.

$$-6ad+9d^{2}=0$$

Subtract 9d^{2} from both sides. Anything subtracted from zero gives its negation.

$$-6ad=-9d^{2}$$

The equation is in standard form.

$$\left(-6d\right)a=-9d^{2}$$

Divide both sides by -6d.

$$\frac{\left(-6d\right)a}{-6d}=-\frac{9d^{2}}{-6d}$$

Dividing by -6d undoes the multiplication by -6d.

$$a=-\frac{9d^{2}}{-6d}$$

Divide -9d^{2} by -6d.

$$a=\frac{3d}{2}$$

Solve for d

$$d=\frac{2a}{3}$$

$$d=0$$