Solve (60-3x/4)^2+x^2=13^2

$$( \frac { 60 - 3 x } { 4 } ) ^ { 2 } + x ^ { 2 } = 13 ^ { 2 }$$

Solve for x

$$x = \frac{16}{5} = 3\frac{1}{5} = 3.2$$

$$x = \frac{56}{5} = 11\frac{1}{5} = 11.2$$

Steps Using the Quadratic Formula

To raise \frac{60-3x}{4} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(60-3x\right)^{2}}{4^{2}}+x^{2}=13^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{2} times \frac{4^{2}}{4^{2}}.

$$\frac{\left(60-3x\right)^{2}}{4^{2}}+\frac{x^{2}\times 4^{2}}{4^{2}}=13^{2}$$

Since \frac{\left(60-3x\right)^{2}}{4^{2}} and \frac{x^{2}\times 4^{2}}{4^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(60-3x\right)^{2}+x^{2}\times 4^{2}}{4^{2}}=13^{2}$$

Do the multiplications in \left(60-3x\right)^{2}+x^{2}\times 4^{2}.

$$\frac{3600-360x+9x^{2}+16x^{2}}{4^{2}}=13^{2}$$

Combine like terms in 3600-360x+9x^{2}+16x^{2}.

$$\frac{3600-360x+25x^{2}}{4^{2}}=13^{2}$$

Calculate 13 to the power of 2 and get 169.

$$\frac{3600-360x+25x^{2}}{4^{2}}=169$$

Calculate 4 to the power of 2 and get 16.

$$\frac{3600-360x+25x^{2}}{16}=169$$

Divide each term of 3600-360x+25x^{2} by 16 to get 225-\frac{45}{2}x+\frac{25}{16}x^{2}.

$$225-\frac{45}{2}x+\frac{25}{16}x^{2}=169$$

Subtract 169 from both sides.

$$225-\frac{45}{2}x+\frac{25}{16}x^{2}-169=0$$

Subtract 169 from 225 to get 56.

$$56-\frac{45}{2}x+\frac{25}{16}x^{2}=0$$

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.

$$\frac{25}{16}x^{2}-\frac{45}{2}x+56=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{25}{16} for a, -\frac{45}{2} for b, and 56 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{45}{2}\right)±\sqrt{\left(-\frac{45}{2}\right)^{2}-4\times \frac{25}{16}\times 56}}{2\times \frac{25}{16}}$$

Square -\frac{45}{2} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{45}{2}\right)±\sqrt{\frac{2025}{4}-4\times \frac{25}{16}\times 56}}{2\times \frac{25}{16}}$$

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

$$x=\frac{-\left(-\frac{45}{2}\right)±\sqrt{\frac{2025}{4}-\frac{25}{4}\times 56}}{2\times \frac{25}{16}}$$

Multiply -\frac{25}{4} times 56.

$$x=\frac{-\left(-\frac{45}{2}\right)±\sqrt{\frac{2025}{4}-350}}{2\times \frac{25}{16}}$$

Add \frac{2025}{4} to -350.

$$x=\frac{-\left(-\frac{45}{2}\right)±\sqrt{\frac{625}{4}}}{2\times \frac{25}{16}}$$

Take the square root of \frac{625}{4}.

$$x=\frac{-\left(-\frac{45}{2}\right)±\frac{25}{2}}{2\times \frac{25}{16}}$$

The opposite of -\frac{45}{2} is \frac{45}{2}.

$$x=\frac{\frac{45}{2}±\frac{25}{2}}{2\times \frac{25}{16}}$$

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

$$x=\frac{\frac{45}{2}±\frac{25}{2}}{\frac{25}{8}}$$

Now solve the equation x=\frac{\frac{45}{2}±\frac{25}{2}}{\frac{25}{8}} when ± is plus. Add \frac{45}{2} to \frac{25}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{35}{\frac{25}{8}}$$

Divide 35 by \frac{25}{8} by multiplying 35 by the reciprocal of \frac{25}{8}.

$$x=\frac{56}{5}$$

Now solve the equation x=\frac{\frac{45}{2}±\frac{25}{2}}{\frac{25}{8}} when ± is minus. Subtract \frac{25}{2} from \frac{45}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{10}{\frac{25}{8}}$$

Divide 10 by \frac{25}{8} by multiplying 10 by the reciprocal of \frac{25}{8}.

$$x=\frac{16}{5}$$

The equation is now solved.

$$x=\frac{56}{5} x=\frac{16}{5}$$

Steps for Completing the Square

To raise \frac{60-3x}{4} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(60-3x\right)^{2}}{4^{2}}+x^{2}=13^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{2} times \frac{4^{2}}{4^{2}}.

$$\frac{\left(60-3x\right)^{2}}{4^{2}}+\frac{x^{2}\times 4^{2}}{4^{2}}=13^{2}$$

Since \frac{\left(60-3x\right)^{2}}{4^{2}} and \frac{x^{2}\times 4^{2}}{4^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(60-3x\right)^{2}+x^{2}\times 4^{2}}{4^{2}}=13^{2}$$

Do the multiplications in \left(60-3x\right)^{2}+x^{2}\times 4^{2}.

$$\frac{3600-360x+9x^{2}+16x^{2}}{4^{2}}=13^{2}$$

Combine like terms in 3600-360x+9x^{2}+16x^{2}.

$$\frac{3600-360x+25x^{2}}{4^{2}}=13^{2}$$

Calculate 13 to the power of 2 and get 169.

$$\frac{3600-360x+25x^{2}}{4^{2}}=169$$

Calculate 4 to the power of 2 and get 16.

$$\frac{3600-360x+25x^{2}}{16}=169$$

Divide each term of 3600-360x+25x^{2} by 16 to get 225-\frac{45}{2}x+\frac{25}{16}x^{2}.

$$225-\frac{45}{2}x+\frac{25}{16}x^{2}=169$$

Subtract 225 from both sides.

$$-\frac{45}{2}x+\frac{25}{16}x^{2}=169-225$$

Subtract 225 from 169 to get -56.

$$-\frac{45}{2}x+\frac{25}{16}x^{2}=-56$$

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.

$$\frac{25}{16}x^{2}-\frac{45}{2}x=-56$$

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

$$\frac{\frac{25}{16}x^{2}-\frac{45}{2}x}{\frac{25}{16}}=-\frac{56}{\frac{25}{16}}$$

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

$$x^{2}+\left(-\frac{\frac{45}{2}}{\frac{25}{16}}\right)x=-\frac{56}{\frac{25}{16}}$$

Divide -\frac{45}{2} by \frac{25}{16} by multiplying -\frac{45}{2} by the reciprocal of \frac{25}{16}.

$$x^{2}-\frac{72}{5}x=-\frac{56}{\frac{25}{16}}$$

Divide -56 by \frac{25}{16} by multiplying -56 by the reciprocal of \frac{25}{16}.

$$x^{2}-\frac{72}{5}x=-\frac{896}{25}$$

Divide -\frac{72}{5}, the coefficient of the x term, by 2 to get -\frac{36}{5}. Then add the square of -\frac{36}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{72}{5}x+\left(-\frac{36}{5}\right)^{2}=-\frac{896}{25}+\left(-\frac{36}{5}\right)^{2}$$

Square -\frac{36}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{72}{5}x+\frac{1296}{25}=\frac{-896+1296}{25}$$

Add -\frac{896}{25} to \frac{1296}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{72}{5}x+\frac{1296}{25}=16$$

Factor x^{2}-\frac{72}{5}x+\frac{1296}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{36}{5}\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{36}{5}\right)^{2}}=\sqrt{16}$$

Simplify.

$$x-\frac{36}{5}=4 x-\frac{36}{5}=-4$$

Add \frac{36}{5} to both sides of the equation.

$$x=\frac{56}{5} x=\frac{16}{5}$$