Solve (a+10)^2=(a-d+10)(a+d+11)

$$( a + 10 ) ^ { 2 } = ( a - d + 10 ) ( a + d + 11 )$$

Solve for a

$$a=d^{2}+d-10$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(a+10\right)^{2}.

$$a^{2}+20a+100=\left(a-d+10\right)\left(a+d+11\right)$$

Use the distributive property to multiply a-d+10 by a+d+11 and combine like terms.

$$a^{2}+20a+100=a^{2}+21a-d^{2}-d+110$$

Subtract a^{2} from both sides.

$$a^{2}+20a+100-a^{2}=21a-d^{2}-d+110$$

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

$$20a+100=21a-d^{2}-d+110$$

Subtract 21a from both sides.

$$20a+100-21a=-d^{2}-d+110$$

Combine 20a and -21a to get -a.

$$-a+100=-d^{2}-d+110$$

Subtract 100 from both sides.

$$-a=-d^{2}-d+110-100$$

Subtract 100 from 110 to get 10.

$$-a=-d^{2}-d+10$$

The equation is in standard form.

$$-a=10-d-d^{2}$$

Divide both sides by -1.

$$\frac{-a}{-1}=\frac{10-d-d^{2}}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$a=\frac{10-d-d^{2}}{-1}$$

Divide -d^{2}-d+10 by -1.

$$a=d^{2}+d-10$$

Solve for d (complex solution)

$$d=\frac{\sqrt{4a+41}-1}{2}$$

$$d=\frac{-\sqrt{4a+41}-1}{2}$$

Solve for d

$$d=\frac{\sqrt{4a+41}-1}{2}$$

$$d=\frac{-\sqrt{4a+41}-1}{2}\text{, }a\geq -\frac{41}{4}$$