Solve {left({a}_{1}+4xright)}^2=left({a}_{1}+xright)left({a}_{1}+10xright)

$${ \left( { a }_{ 1 } +4x \right) }^{ 2 } = \left( { a }_{ 1 } +x \right) \left( { a }_{ 1 } +10x \right)$$

Solve for a_1 (complex solution)

$$\left\{\begin{matrix}\\a_{1}=2x\text{, }&\text{unconditionally}\\a_{1}\in \mathrm{C}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(a_{1}+4x\right)^{2}.

$$a_{1}^{2}+8a_{1}x+16x^{2}=\left(a_{1}+x\right)\left(a_{1}+10x\right)$$

Use the distributive property to multiply a_{1}+x by a_{1}+10x and combine like terms.

$$a_{1}^{2}+8a_{1}x+16x^{2}=a_{1}^{2}+11a_{1}x+10x^{2}$$

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

$$a_{1}^{2}+8a_{1}x+16x^{2}-a_{1}^{2}=11a_{1}x+10x^{2}$$

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

$$8a_{1}x+16x^{2}=11a_{1}x+10x^{2}$$

Subtract 11a_{1}x from both sides.

$$8a_{1}x+16x^{2}-11a_{1}x=10x^{2}$$

Combine 8a_{1}x and -11a_{1}x to get -3a_{1}x.

$$-3a_{1}x+16x^{2}=10x^{2}$$

Subtract 16x^{2} from both sides.

$$-3a_{1}x=10x^{2}-16x^{2}$$

Combine 10x^{2} and -16x^{2} to get -6x^{2}.

$$-3a_{1}x=-6x^{2}$$

The equation is in standard form.

$$\left(-3x\right)a_{1}=-6x^{2}$$

Divide both sides by -3x.

$$\frac{\left(-3x\right)a_{1}}{-3x}=-\frac{6x^{2}}{-3x}$$

Dividing by -3x undoes the multiplication by -3x.

$$a_{1}=-\frac{6x^{2}}{-3x}$$

Divide -6x^{2} by -3x.

$$a_{1}=2x$$

Solve for a_1

$$\left\{\begin{matrix}\\a_{1}=2x\text{, }&\text{unconditionally}\\a_{1}\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(a_{1}+4x\right)^{2}.

$$a_{1}^{2}+8a_{1}x+16x^{2}=\left(a_{1}+x\right)\left(a_{1}+10x\right)$$

Use the distributive property to multiply a_{1}+x by a_{1}+10x and combine like terms.

$$a_{1}^{2}+8a_{1}x+16x^{2}=a_{1}^{2}+11a_{1}x+10x^{2}$$

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

$$a_{1}^{2}+8a_{1}x+16x^{2}-a_{1}^{2}=11a_{1}x+10x^{2}$$

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

$$8a_{1}x+16x^{2}=11a_{1}x+10x^{2}$$

Subtract 11a_{1}x from both sides.

$$8a_{1}x+16x^{2}-11a_{1}x=10x^{2}$$

Combine 8a_{1}x and -11a_{1}x to get -3a_{1}x.

$$-3a_{1}x+16x^{2}=10x^{2}$$

Subtract 16x^{2} from both sides.

$$-3a_{1}x=10x^{2}-16x^{2}$$

Combine 10x^{2} and -16x^{2} to get -6x^{2}.

$$-3a_{1}x=-6x^{2}$$

The equation is in standard form.

$$\left(-3x\right)a_{1}=-6x^{2}$$

Divide both sides by -3x.

$$\frac{\left(-3x\right)a_{1}}{-3x}=-\frac{6x^{2}}{-3x}$$

Dividing by -3x undoes the multiplication by -3x.

$$a_{1}=-\frac{6x^{2}}{-3x}$$

Divide -6x^{2} by -3x.

$$a_{1}=2x$$

Solve for x

$$x=\frac{a_{1}}{2}$$

$$x=0$$