Solve 5r^2-44r+120=-30

$$5 r ^ { 2 } - 44 r + 120 = - 30$$

Solve for r

$$r=\frac{22+\sqrt{266}i}{5}\approx 4.4+3.261901286i$$

$$r=\frac{-\sqrt{266}i+22}{5}\approx 4.4-3.261901286i$$

Steps Using the Quadratic Formula

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.

$$5r^{2}-44r+120=-30$$

Add 30 to both sides of the equation.

$$5r^{2}-44r+120-\left(-30\right)=-30-\left(-30\right)$$

Subtracting -30 from itself leaves 0.

$$5r^{2}-44r+120-\left(-30\right)=0$$

Subtract -30 from 120.

$$5r^{2}-44r+150=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -44 for b, and 150 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-\left(-44\right)±\sqrt{\left(-44\right)^{2}-4\times 5\times 150}}{2\times 5}$$

Square -44.

$$r=\frac{-\left(-44\right)±\sqrt{1936-4\times 5\times 150}}{2\times 5}$$

Multiply -4 times 5.

$$r=\frac{-\left(-44\right)±\sqrt{1936-20\times 150}}{2\times 5}$$

Multiply -20 times 150.

$$r=\frac{-\left(-44\right)±\sqrt{1936-3000}}{2\times 5}$$

Add 1936 to -3000.

$$r=\frac{-\left(-44\right)±\sqrt{-1064}}{2\times 5}$$

Take the square root of -1064.

$$r=\frac{-\left(-44\right)±2\sqrt{266}i}{2\times 5}$$

The opposite of -44 is 44.

$$r=\frac{44±2\sqrt{266}i}{2\times 5}$$

Multiply 2 times 5.

$$r=\frac{44±2\sqrt{266}i}{10}$$

Now solve the equation r=\frac{44±2\sqrt{266}i}{10} when ± is plus. Add 44 to 2i\sqrt{266}.

$$r=\frac{44+2\sqrt{266}i}{10}$$

Divide 44+2i\sqrt{266} by 10.

$$r=\frac{22+\sqrt{266}i}{5}$$

Now solve the equation r=\frac{44±2\sqrt{266}i}{10} when ± is minus. Subtract 2i\sqrt{266} from 44.

$$r=\frac{-2\sqrt{266}i+44}{10}$$

Divide 44-2i\sqrt{266} by 10.

$$r=\frac{-\sqrt{266}i+22}{5}$$

The equation is now solved.

$$r=\frac{22+\sqrt{266}i}{5} r=\frac{-\sqrt{266}i+22}{5}$$

Steps for Completing the Square

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.

$$5r^{2}-44r+120=-30$$

Subtract 120 from both sides of the equation.

$$5r^{2}-44r+120-120=-30-120$$

Subtracting 120 from itself leaves 0.

$$5r^{2}-44r=-30-120$$

Subtract 120 from -30.

$$5r^{2}-44r=-150$$

Divide both sides by 5.

$$\frac{5r^{2}-44r}{5}=-\frac{150}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$r^{2}-\frac{44}{5}r=-\frac{150}{5}$$

Divide -150 by 5.

$$r^{2}-\frac{44}{5}r=-30$$

Divide -\frac{44}{5}, the coefficient of the x term, by 2 to get -\frac{22}{5}. Then add the square of -\frac{22}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$r^{2}-\frac{44}{5}r+\left(-\frac{22}{5}\right)^{2}=-30+\left(-\frac{22}{5}\right)^{2}$$

Square -\frac{22}{5} by squaring both the numerator and the denominator of the fraction.

$$r^{2}-\frac{44}{5}r+\frac{484}{25}=-30+\frac{484}{25}$$

Add -30 to \frac{484}{25}.

$$r^{2}-\frac{44}{5}r+\frac{484}{25}=-\frac{266}{25}$$

Factor r^{2}-\frac{44}{5}r+\frac{484}{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}.

$$\left(r-\frac{22}{5}\right)^{2}=-\frac{266}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r-\frac{22}{5}\right)^{2}}=\sqrt{-\frac{266}{25}}$$

Simplify.

$$r-\frac{22}{5}=\frac{\sqrt{266}i}{5} r-\frac{22}{5}=-\frac{\sqrt{266}i}{5}$$

Add \frac{22}{5} to both sides of the equation.

$$r=\frac{22+\sqrt{266}i}{5} r=\frac{-\sqrt{266}i+22}{5}$$