Solve 49+14r+r^2+30r/4=4r-1000

$$49 + 14 r + r ^ { 2 } + \frac { 30 r } { 4 } = 4 r - 1000$$

Solve for r

$$r=\frac{-35+\sqrt{15559}i}{4}\approx -8.75+31.183930156i$$

$$r=\frac{-\sqrt{15559}i-35}{4}\approx -8.75-31.183930156i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 4.

$$196+56r+4r^{2}+30r=16r-4000$$

Combine 56r and 30r to get 86r.

$$196+86r+4r^{2}=16r-4000$$

Subtract 16r from both sides.

$$196+86r+4r^{2}-16r=-4000$$

Combine 86r and -16r to get 70r.

$$196+70r+4r^{2}=-4000$$

Add 4000 to both sides.

$$196+70r+4r^{2}+4000=0$$

Add 196 and 4000 to get 4196.

$$4196+70r+4r^{2}=0$$

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.

$$4r^{2}+70r+4196=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 70 for b, and 4196 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-70±\sqrt{70^{2}-4\times 4\times 4196}}{2\times 4}$$

Square 70.

$$r=\frac{-70±\sqrt{4900-4\times 4\times 4196}}{2\times 4}$$

Multiply -4 times 4.

$$r=\frac{-70±\sqrt{4900-16\times 4196}}{2\times 4}$$

Multiply -16 times 4196.

$$r=\frac{-70±\sqrt{4900-67136}}{2\times 4}$$

Add 4900 to -67136.

$$r=\frac{-70±\sqrt{-62236}}{2\times 4}$$

Take the square root of -62236.

$$r=\frac{-70±2\sqrt{15559}i}{2\times 4}$$

Multiply 2 times 4.

$$r=\frac{-70±2\sqrt{15559}i}{8}$$

Now solve the equation r=\frac{-70±2\sqrt{15559}i}{8} when ± is plus. Add -70 to 2i\sqrt{15559}.

$$r=\frac{-70+2\sqrt{15559}i}{8}$$

Divide -70+2i\sqrt{15559} by 8.

$$r=\frac{-35+\sqrt{15559}i}{4}$$

Now solve the equation r=\frac{-70±2\sqrt{15559}i}{8} when ± is minus. Subtract 2i\sqrt{15559} from -70.

$$r=\frac{-2\sqrt{15559}i-70}{8}$$

Divide -70-2i\sqrt{15559} by 8.

$$r=\frac{-\sqrt{15559}i-35}{4}$$

The equation is now solved.

$$r=\frac{-35+\sqrt{15559}i}{4} r=\frac{-\sqrt{15559}i-35}{4}$$

Steps for Completing the Square

Multiply both sides of the equation by 4.

$$196+56r+4r^{2}+30r=16r-4000$$

Combine 56r and 30r to get 86r.

$$196+86r+4r^{2}=16r-4000$$

Subtract 16r from both sides.

$$196+86r+4r^{2}-16r=-4000$$

Combine 86r and -16r to get 70r.

$$196+70r+4r^{2}=-4000$$

Subtract 196 from both sides.

$$70r+4r^{2}=-4000-196$$

Subtract 196 from -4000 to get -4196.

$$70r+4r^{2}=-4196$$

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.

$$4r^{2}+70r=-4196$$

Divide both sides by 4.

$$\frac{4r^{2}+70r}{4}=-\frac{4196}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$r^{2}+\frac{70}{4}r=-\frac{4196}{4}$$

Reduce the fraction \frac{70}{4} to lowest terms by extracting and canceling out 2.

$$r^{2}+\frac{35}{2}r=-\frac{4196}{4}$$

Divide -4196 by 4.

$$r^{2}+\frac{35}{2}r=-1049$$

Divide \frac{35}{2}, the coefficient of the x term, by 2 to get \frac{35}{4}. Then add the square of \frac{35}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$r^{2}+\frac{35}{2}r+\left(\frac{35}{4}\right)^{2}=-1049+\left(\frac{35}{4}\right)^{2}$$

Square \frac{35}{4} by squaring both the numerator and the denominator of the fraction.

$$r^{2}+\frac{35}{2}r+\frac{1225}{16}=-1049+\frac{1225}{16}$$

Add -1049 to \frac{1225}{16}.

$$r^{2}+\frac{35}{2}r+\frac{1225}{16}=-\frac{15559}{16}$$

Factor r^{2}+\frac{35}{2}r+\frac{1225}{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}.

$$\left(r+\frac{35}{4}\right)^{2}=-\frac{15559}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r+\frac{35}{4}\right)^{2}}=\sqrt{-\frac{15559}{16}}$$

Simplify.

$$r+\frac{35}{4}=\frac{\sqrt{15559}i}{4} r+\frac{35}{4}=-\frac{\sqrt{15559}i}{4}$$

Subtract \frac{35}{4} from both sides of the equation.

$$r=\frac{-35+\sqrt{15559}i}{4} r=\frac{-\sqrt{15559}i-35}{4}$$