Solve 13/48+x/2-2x+1/6=1-(1/4-x)(x+frac{1}{4})-(x+frac{1}{2})^2

$$\frac { 13 } { 48 } + \frac { x } { 2 } - \frac { 2 x + 1 } { 6 } = 1 - ( \frac { 1 } { 4 } - x ) ( x + \frac { 1 } { 4 } ) - ( x + \frac { 1 } { 2 } ) ^ { 2 }$$

Solve for x

$$x=\frac{1}{2}=0.5$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 48, the least common multiple of 48,2,6,4.

$$13+24x-8\left(2x+1\right)=48\left(1-\left(\frac{1}{4}-x\right)\left(x+\frac{1}{4}\right)\right)-48\left(x+\frac{1}{2}\right)^{2}$$

Use the distributive property to multiply -8 by 2x+1.

$$13+24x-16x-8=48\left(1-\left(\frac{1}{4}-x\right)\left(x+\frac{1}{4}\right)\right)-48\left(x+\frac{1}{2}\right)^{2}$$

Combine 24x and -16x to get 8x.

$$13+8x-8=48\left(1-\left(\frac{1}{4}-x\right)\left(x+\frac{1}{4}\right)\right)-48\left(x+\frac{1}{2}\right)^{2}$$

Subtract 8 from 13 to get 5.

$$5+8x=48\left(1-\left(\frac{1}{4}-x\right)\left(x+\frac{1}{4}\right)\right)-48\left(x+\frac{1}{2}\right)^{2}$$

Consider \left(\frac{1}{4}-x\right)\left(x+\frac{1}{4}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square \frac{1}{4}.

$$5+8x=48\left(1-\left(\frac{1}{16}-x^{2}\right)\right)-48\left(x+\frac{1}{2}\right)^{2}$$

To find the opposite of \frac{1}{16}-x^{2}, find the opposite of each term.

$$5+8x=48\left(1-\frac{1}{16}+x^{2}\right)-48\left(x+\frac{1}{2}\right)^{2}$$

Subtract \frac{1}{16} from 1 to get \frac{15}{16}.

$$5+8x=48\left(\frac{15}{16}+x^{2}\right)-48\left(x+\frac{1}{2}\right)^{2}$$

Use the distributive property to multiply 48 by \frac{15}{16}+x^{2}.

$$5+8x=45+48x^{2}-48\left(x+\frac{1}{2}\right)^{2}$$

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

$$5+8x=45+48x^{2}-48\left(x^{2}+x+\frac{1}{4}\right)$$

Use the distributive property to multiply -48 by x^{2}+x+\frac{1}{4}.

$$5+8x=45+48x^{2}-48x^{2}-48x-12$$

Combine 48x^{2} and -48x^{2} to get 0.

$$5+8x=45-48x-12$$

Subtract 12 from 45 to get 33.

$$5+8x=33-48x$$

Add 48x to both sides.

$$5+8x+48x=33$$

Combine 8x and 48x to get 56x.

$$5+56x=33$$

Subtract 5 from both sides.

$$56x=33-5$$

Subtract 5 from 33 to get 28.

$$56x=28$$

Divide both sides by 56.

$$x=\frac{28}{56}$$

Reduce the fraction \frac{28}{56} to lowest terms by extracting and canceling out 28.

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