Solve (60+56x)^2=64+49+112x

$$( 60 + 56 x ) ^ { 2 } = 64 + 49 + 112 x$$

Solve for x (complex solution)

$$x=\frac{-59+\sqrt{6}i}{56}\approx -1.053571429+0.043740888i$$

$$x=\frac{-\sqrt{6}i-59}{56}\approx -1.053571429-0.043740888i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(60+56x\right)^{2}.

$$3600+6720x+3136x^{2}=64+49+112x$$

Add 64 and 49 to get 113.

$$3600+6720x+3136x^{2}=113+112x$$

Subtract 113 from both sides.

$$3600+6720x+3136x^{2}-113=112x$$

Subtract 113 from 3600 to get 3487.

$$3487+6720x+3136x^{2}=112x$$

Subtract 112x from both sides.

$$3487+6720x+3136x^{2}-112x=0$$

Combine 6720x and -112x to get 6608x.

$$3487+6608x+3136x^{2}=0$$

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.

$$3136x^{2}+6608x+3487=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3136 for a, 6608 for b, and 3487 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6608±\sqrt{6608^{2}-4\times 3136\times 3487}}{2\times 3136}$$

Square 6608.

$$x=\frac{-6608±\sqrt{43665664-4\times 3136\times 3487}}{2\times 3136}$$

Multiply -4 times 3136.

$$x=\frac{-6608±\sqrt{43665664-12544\times 3487}}{2\times 3136}$$

Multiply -12544 times 3487.

$$x=\frac{-6608±\sqrt{43665664-43740928}}{2\times 3136}$$

Add 43665664 to -43740928.

$$x=\frac{-6608±\sqrt{-75264}}{2\times 3136}$$

Take the square root of -75264.

$$x=\frac{-6608±112\sqrt{6}i}{2\times 3136}$$

Multiply 2 times 3136.

$$x=\frac{-6608±112\sqrt{6}i}{6272}$$

Now solve the equation x=\frac{-6608±112\sqrt{6}i}{6272} when ± is plus. Add -6608 to 112i\sqrt{6}.

$$x=\frac{-6608+112\sqrt{6}i}{6272}$$

Divide -6608+112i\sqrt{6} by 6272.

$$x=\frac{-59+\sqrt{6}i}{56}$$

Now solve the equation x=\frac{-6608±112\sqrt{6}i}{6272} when ± is minus. Subtract 112i\sqrt{6} from -6608.

$$x=\frac{-112\sqrt{6}i-6608}{6272}$$

Divide -6608-112i\sqrt{6} by 6272.

$$x=\frac{-\sqrt{6}i-59}{56}$$

The equation is now solved.

$$x=\frac{-59+\sqrt{6}i}{56} x=\frac{-\sqrt{6}i-59}{56}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(60+56x\right)^{2}.

$$3600+6720x+3136x^{2}=64+49+112x$$

Add 64 and 49 to get 113.

$$3600+6720x+3136x^{2}=113+112x$$

Subtract 112x from both sides.

$$3600+6720x+3136x^{2}-112x=113$$

Combine 6720x and -112x to get 6608x.

$$3600+6608x+3136x^{2}=113$$

Subtract 3600 from both sides.

$$6608x+3136x^{2}=113-3600$$

Subtract 3600 from 113 to get -3487.

$$6608x+3136x^{2}=-3487$$

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.

$$3136x^{2}+6608x=-3487$$

Divide both sides by 3136.

$$\frac{3136x^{2}+6608x}{3136}=-\frac{3487}{3136}$$

Dividing by 3136 undoes the multiplication by 3136.

$$x^{2}+\frac{6608}{3136}x=-\frac{3487}{3136}$$

Reduce the fraction \frac{6608}{3136} to lowest terms by extracting and canceling out 112.

$$x^{2}+\frac{59}{28}x=-\frac{3487}{3136}$$

Divide \frac{59}{28}, the coefficient of the x term, by 2 to get \frac{59}{56}. Then add the square of \frac{59}{56} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{59}{28}x+\left(\frac{59}{56}\right)^{2}=-\frac{3487}{3136}+\left(\frac{59}{56}\right)^{2}$$

Square \frac{59}{56} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{59}{28}x+\frac{3481}{3136}=\frac{-3487+3481}{3136}$$

Add -\frac{3487}{3136} to \frac{3481}{3136} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{59}{28}x+\frac{3481}{3136}=-\frac{3}{1568}$$

Factor x^{2}+\frac{59}{28}x+\frac{3481}{3136}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{59}{56}\right)^{2}=-\frac{3}{1568}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{59}{56}\right)^{2}}=\sqrt{-\frac{3}{1568}}$$

Simplify.

$$x+\frac{59}{56}=\frac{\sqrt{6}i}{56} x+\frac{59}{56}=-\frac{\sqrt{6}i}{56}$$

Subtract \frac{59}{56} from both sides of the equation.

$$x=\frac{-59+\sqrt{6}i}{56} x=\frac{-\sqrt{6}i-59}{56}$$