Solve (3x-7)(3x+7)/6=(x-2)^2/2+(2x+1)^2/4

$$\frac { ( 3 x - 7 ) ( 3 x + 7 ) } { 6 } = \frac { ( x - 2 ) ^ { 2 } } { 2 } + \frac { ( 2 x + 1 ) ^ { 2 } } { 4 }$$

Solve for x

$$x = \frac{125}{12} = 10\frac{5}{12} \approx 10.416666667$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 12, the least common multiple of 6,2,4.

$$2\left(3x-7\right)\left(3x+7\right)=6\left(x-2\right)^{2}+3\left(2x+1\right)^{2}$$

Use the distributive property to multiply 2 by 3x-7.

$$\left(6x-14\right)\left(3x+7\right)=6\left(x-2\right)^{2}+3\left(2x+1\right)^{2}$$

Use the distributive property to multiply 6x-14 by 3x+7 and combine like terms.

$$18x^{2}-98=6\left(x-2\right)^{2}+3\left(2x+1\right)^{2}$$

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

$$18x^{2}-98=6\left(x^{2}-4x+4\right)+3\left(2x+1\right)^{2}$$

Use the distributive property to multiply 6 by x^{2}-4x+4.

$$18x^{2}-98=6x^{2}-24x+24+3\left(2x+1\right)^{2}$$

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

$$18x^{2}-98=6x^{2}-24x+24+3\left(4x^{2}+4x+1\right)$$

Use the distributive property to multiply 3 by 4x^{2}+4x+1.

$$18x^{2}-98=6x^{2}-24x+24+12x^{2}+12x+3$$

Combine 6x^{2} and 12x^{2} to get 18x^{2}.

$$18x^{2}-98=18x^{2}-24x+24+12x+3$$

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

$$18x^{2}-98=18x^{2}-12x+24+3$$

Add 24 and 3 to get 27.

$$18x^{2}-98=18x^{2}-12x+27$$

Subtract 18x^{2} from both sides.

$$18x^{2}-98-18x^{2}=-12x+27$$

Combine 18x^{2} and -18x^{2} to get 0.

$$-98=-12x+27$$

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

$$-12x+27=-98$$

Subtract 27 from both sides.

$$-12x=-98-27$$

Subtract 27 from -98 to get -125.

$$-12x=-125$$

Divide both sides by -12.

$$x=\frac{-125}{-12}$$

Fraction \frac{-125}{-12} can be simplified to \frac{125}{12} by removing the negative sign from both the numerator and the denominator.

$$x=\frac{125}{12}$$