Solve (r-2)=r^2+5

$$( r - 2 ) = r ^ { 2 } + 5$$

Solve for r

$$r=\frac{-3\sqrt{3}i+1}{2}\approx 0.5-2.598076211i$$

$$r=\frac{1+3\sqrt{3}i}{2}\approx 0.5+2.598076211i$$

Steps Using the Quadratic Formula

Subtract r^{2} from both sides.

$$r-2-r^{2}=5$$

Subtract 5 from both sides.

$$r-2-r^{2}-5=0$$

Subtract 5 from -2 to get -7.

$$r-7-r^{2}=0$$

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.

$$-r^{2}+r-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 1 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-1±\sqrt{1^{2}-4\left(-1\right)\left(-7\right)}}{2\left(-1\right)}$$

Square 1.

$$r=\frac{-1±\sqrt{1-4\left(-1\right)\left(-7\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$r=\frac{-1±\sqrt{1+4\left(-7\right)}}{2\left(-1\right)}$$

Multiply 4 times -7.

$$r=\frac{-1±\sqrt{1-28}}{2\left(-1\right)}$$

Add 1 to -28.

$$r=\frac{-1±\sqrt{-27}}{2\left(-1\right)}$$

Take the square root of -27.

$$r=\frac{-1±3\sqrt{3}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$r=\frac{-1±3\sqrt{3}i}{-2}$$

Now solve the equation r=\frac{-1±3\sqrt{3}i}{-2} when ± is plus. Add -1 to 3i\sqrt{3}.

$$r=\frac{-1+3\sqrt{3}i}{-2}$$

Divide -1+3i\sqrt{3} by -2.

$$r=\frac{-3\sqrt{3}i+1}{2}$$

Now solve the equation r=\frac{-1±3\sqrt{3}i}{-2} when ± is minus. Subtract 3i\sqrt{3} from -1.

$$r=\frac{-3\sqrt{3}i-1}{-2}$$

Divide -1-3i\sqrt{3} by -2.

$$r=\frac{1+3\sqrt{3}i}{2}$$

The equation is now solved.

$$r=\frac{-3\sqrt{3}i+1}{2} r=\frac{1+3\sqrt{3}i}{2}$$

Steps for Completing the Square

Subtract r^{2} from both sides.

$$r-2-r^{2}=5$$

Add 2 to both sides.

$$r-r^{2}=5+2$$

Add 5 and 2 to get 7.

$$r-r^{2}=7$$

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.

$$-r^{2}+r=7$$

Divide both sides by -1.

$$\frac{-r^{2}+r}{-1}=\frac{7}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$r^{2}+\frac{1}{-1}r=\frac{7}{-1}$$

Divide 1 by -1.

$$r^{2}-r=\frac{7}{-1}$$

Divide 7 by -1.

$$r^{2}-r=-7$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$r^{2}-r+\left(-\frac{1}{2}\right)^{2}=-7+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$r^{2}-r+\frac{1}{4}=-7+\frac{1}{4}$$

Add -7 to \frac{1}{4}.

$$r^{2}-r+\frac{1}{4}=-\frac{27}{4}$$

Factor r^{2}-r+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(r-\frac{1}{2}\right)^{2}=-\frac{27}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r-\frac{1}{2}\right)^{2}}=\sqrt{-\frac{27}{4}}$$

Simplify.

$$r-\frac{1}{2}=\frac{3\sqrt{3}i}{2} r-\frac{1}{2}=-\frac{3\sqrt{3}i}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$r=\frac{1+3\sqrt{3}i}{2} r=\frac{-3\sqrt{3}i+1}{2}$$