Solve left(x-6right)^2+49=left(4-3xright)^2+(x+7)(x+5)-2

$${ \left(x-6 \right) }^{ 2 } +49= { \left(4-3x \right) }^{ 2 } +(x+7)(x+5)-2$$

Solve for x

$$x=2$$

$$x=-2$$

Steps by Finding Square Root

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-6\right)^{2}.

$$x^{2}-12x+36+49=\left(4-3x\right)^{2}+\left(x+7\right)\left(x+5\right)-2$$

Add 36 and 49 to get 85.

$$x^{2}-12x+85=\left(4-3x\right)^{2}+\left(x+7\right)\left(x+5\right)-2$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-3x\right)^{2}.

$$x^{2}-12x+85=16-24x+9x^{2}+\left(x+7\right)\left(x+5\right)-2$$

Use the distributive property to multiply x+7 by x+5 and combine like terms.

$$x^{2}-12x+85=16-24x+9x^{2}+x^{2}+12x+35-2$$

Combine 9x^{2} and x^{2} to get 10x^{2}.

$$x^{2}-12x+85=16-24x+10x^{2}+12x+35-2$$

Combine -24x and 12x to get -12x.

$$x^{2}-12x+85=16-12x+10x^{2}+35-2$$

Add 16 and 35 to get 51.

$$x^{2}-12x+85=51-12x+10x^{2}-2$$

Subtract 2 from 51 to get 49.

$$x^{2}-12x+85=49-12x+10x^{2}$$

Add 12x to both sides.

$$x^{2}-12x+85+12x=49+10x^{2}$$

Combine -12x and 12x to get 0.

$$x^{2}+85=49+10x^{2}$$

Subtract 10x^{2} from both sides.

$$x^{2}+85-10x^{2}=49$$

Combine x^{2} and -10x^{2} to get -9x^{2}.

$$-9x^{2}+85=49$$

Subtract 85 from both sides.

$$-9x^{2}=49-85$$

Subtract 85 from 49 to get -36.

$$-9x^{2}=-36$$

Divide both sides by -9.

$$x^{2}=\frac{-36}{-9}$$

Divide -36 by -9 to get 4.

$$x^{2}=4$$

Take the square root of both sides of the equation.

$$x=2 x=-2$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-6\right)^{2}.

$$x^{2}-12x+36+49=\left(4-3x\right)^{2}+\left(x+7\right)\left(x+5\right)-2$$

Add 36 and 49 to get 85.

$$x^{2}-12x+85=\left(4-3x\right)^{2}+\left(x+7\right)\left(x+5\right)-2$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-3x\right)^{2}.

$$x^{2}-12x+85=16-24x+9x^{2}+\left(x+7\right)\left(x+5\right)-2$$

Use the distributive property to multiply x+7 by x+5 and combine like terms.

$$x^{2}-12x+85=16-24x+9x^{2}+x^{2}+12x+35-2$$

Combine 9x^{2} and x^{2} to get 10x^{2}.

$$x^{2}-12x+85=16-24x+10x^{2}+12x+35-2$$

Combine -24x and 12x to get -12x.

$$x^{2}-12x+85=16-12x+10x^{2}+35-2$$

Add 16 and 35 to get 51.

$$x^{2}-12x+85=51-12x+10x^{2}-2$$

Subtract 2 from 51 to get 49.

$$x^{2}-12x+85=49-12x+10x^{2}$$

Subtract 49 from both sides.

$$x^{2}-12x+85-49=-12x+10x^{2}$$

Subtract 49 from 85 to get 36.

$$x^{2}-12x+36=-12x+10x^{2}$$

Add 12x to both sides.

$$x^{2}-12x+36+12x=10x^{2}$$

Combine -12x and 12x to get 0.

$$x^{2}+36=10x^{2}$$

Subtract 10x^{2} from both sides.

$$x^{2}+36-10x^{2}=0$$

Combine x^{2} and -10x^{2} to get -9x^{2}.

$$-9x^{2}+36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 0 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-9\right)\times 36}}{2\left(-9\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-9\right)\times 36}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{0±\sqrt{36\times 36}}{2\left(-9\right)}$$

Multiply 36 times 36.

$$x=\frac{0±\sqrt{1296}}{2\left(-9\right)}$$

Take the square root of 1296.

$$x=\frac{0±36}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{0±36}{-18}$$

Now solve the equation x=\frac{0±36}{-18} when ± is plus. Divide 36 by -18.

$$x=-2$$

Now solve the equation x=\frac{0±36}{-18} when ± is minus. Divide -36 by -18.

$$x=2$$

The equation is now solved.

$$x=-2 x=2$$