Solve 8^2=(4+x)^2+(3/4x+3)^2

$$8 ^ { 2 } = ( 4 + x ) ^ { 2 } + ( \frac { 3 } { 4 } x + 3 ) ^ { 2 }$$

Solve for x

$$x = -\frac{52}{5} = -10\frac{2}{5} = -10.4$$

$$x = \frac{12}{5} = 2\frac{2}{5} = 2.4$$

Steps Using the Quadratic Formula

Calculate 8 to the power of 2 and get 64.

$$64=\left(4+x\right)^{2}+\left(\frac{3}{4}x+3\right)^{2}$$

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

$$64=16+8x+x^{2}+\left(\frac{3}{4}x+3\right)^{2}$$

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

$$64=16+8x+x^{2}+\frac{9}{16}x^{2}+\frac{9}{2}x+9$$

Combine x^{2} and \frac{9}{16}x^{2} to get \frac{25}{16}x^{2}.

$$64=16+8x+\frac{25}{16}x^{2}+\frac{9}{2}x+9$$

Combine 8x and \frac{9}{2}x to get \frac{25}{2}x.

$$64=16+\frac{25}{2}x+\frac{25}{16}x^{2}+9$$

Add 16 and 9 to get 25.

$$64=25+\frac{25}{2}x+\frac{25}{16}x^{2}$$

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

$$25+\frac{25}{2}x+\frac{25}{16}x^{2}=64$$

Subtract 64 from both sides.

$$25+\frac{25}{2}x+\frac{25}{16}x^{2}-64=0$$

Subtract 64 from 25 to get -39.

$$-39+\frac{25}{2}x+\frac{25}{16}x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$\frac{25}{16}x^{2}+\frac{25}{2}x-39=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{25}{16} for a, \frac{25}{2} for b, and -39 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{25}{2}±\sqrt{\left(\frac{25}{2}\right)^{2}-4\times \frac{25}{16}\left(-39\right)}}{2\times \frac{25}{16}}$$

Square \frac{25}{2} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{25}{2}±\sqrt{\frac{625}{4}-4\times \frac{25}{16}\left(-39\right)}}{2\times \frac{25}{16}}$$

Multiply -4 times \frac{25}{16}.

$$x=\frac{-\frac{25}{2}±\sqrt{\frac{625}{4}-\frac{25}{4}\left(-39\right)}}{2\times \frac{25}{16}}$$

Multiply -\frac{25}{4} times -39.

$$x=\frac{-\frac{25}{2}±\sqrt{\frac{625+975}{4}}}{2\times \frac{25}{16}}$$

Add \frac{625}{4} to \frac{975}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{25}{2}±\sqrt{400}}{2\times \frac{25}{16}}$$

Take the square root of 400.

$$x=\frac{-\frac{25}{2}±20}{2\times \frac{25}{16}}$$

Multiply 2 times \frac{25}{16}.

$$x=\frac{-\frac{25}{2}±20}{\frac{25}{8}}$$

Now solve the equation x=\frac{-\frac{25}{2}±20}{\frac{25}{8}} when ± is plus. Add -\frac{25}{2} to 20.

$$x=\frac{\frac{15}{2}}{\frac{25}{8}}$$

Divide \frac{15}{2} by \frac{25}{8} by multiplying \frac{15}{2} by the reciprocal of \frac{25}{8}.

$$x=\frac{12}{5}$$

Now solve the equation x=\frac{-\frac{25}{2}±20}{\frac{25}{8}} when ± is minus. Subtract 20 from -\frac{25}{2}.

$$x=-\frac{\frac{65}{2}}{\frac{25}{8}}$$

Divide -\frac{65}{2} by \frac{25}{8} by multiplying -\frac{65}{2} by the reciprocal of \frac{25}{8}.

$$x=-\frac{52}{5}$$

The equation is now solved.

$$x=\frac{12}{5} x=-\frac{52}{5}$$

Steps for Completing the Square

Calculate 8 to the power of 2 and get 64.

$$64=\left(4+x\right)^{2}+\left(\frac{3}{4}x+3\right)^{2}$$

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

$$64=16+8x+x^{2}+\left(\frac{3}{4}x+3\right)^{2}$$

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

$$64=16+8x+x^{2}+\frac{9}{16}x^{2}+\frac{9}{2}x+9$$

Combine x^{2} and \frac{9}{16}x^{2} to get \frac{25}{16}x^{2}.

$$64=16+8x+\frac{25}{16}x^{2}+\frac{9}{2}x+9$$

Combine 8x and \frac{9}{2}x to get \frac{25}{2}x.

$$64=16+\frac{25}{2}x+\frac{25}{16}x^{2}+9$$

Add 16 and 9 to get 25.

$$64=25+\frac{25}{2}x+\frac{25}{16}x^{2}$$

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

$$25+\frac{25}{2}x+\frac{25}{16}x^{2}=64$$

Subtract 25 from both sides.

$$\frac{25}{2}x+\frac{25}{16}x^{2}=64-25$$

Subtract 25 from 64 to get 39.

$$\frac{25}{2}x+\frac{25}{16}x^{2}=39$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$\frac{25}{16}x^{2}+\frac{25}{2}x=39$$

Divide both sides of the equation by \frac{25}{16}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{25}{16}x^{2}+\frac{25}{2}x}{\frac{25}{16}}=\frac{39}{\frac{25}{16}}$$

Dividing by \frac{25}{16} undoes the multiplication by \frac{25}{16}.

$$x^{2}+\frac{\frac{25}{2}}{\frac{25}{16}}x=\frac{39}{\frac{25}{16}}$$

Divide \frac{25}{2} by \frac{25}{16} by multiplying \frac{25}{2} by the reciprocal of \frac{25}{16}.

$$x^{2}+8x=\frac{39}{\frac{25}{16}}$$

Divide 39 by \frac{25}{16} by multiplying 39 by the reciprocal of \frac{25}{16}.

$$x^{2}+8x=\frac{624}{25}$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+8x+4^{2}=\frac{624}{25}+4^{2}$$

Square 4.

$$x^{2}+8x+16=\frac{624}{25}+16$$

Add \frac{624}{25} to 16.

$$x^{2}+8x+16=\frac{1024}{25}$$

Factor x^{2}+8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+4\right)^{2}=\frac{1024}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{\frac{1024}{25}}$$

Simplify.

$$x+4=\frac{32}{5} x+4=-\frac{32}{5}$$

Subtract 4 from both sides of the equation.

$$x=\frac{12}{5} x=-\frac{52}{5}$$