Solve 243h^2+17h=-10

$$243 h ^ { 2 } + 17 h = - 10$$

Solve for h

$$h=\frac{-17+\sqrt{9431}i}{486}\approx -0.034979424+0.199821679i$$

$$h=\frac{-\sqrt{9431}i-17}{486}\approx -0.034979424-0.199821679i$$

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.

$$243h^{2}+17h=-10$$

Add 10 to both sides of the equation.

$$243h^{2}+17h-\left(-10\right)=-10-\left(-10\right)$$

Subtracting -10 from itself leaves 0.

$$243h^{2}+17h-\left(-10\right)=0$$

Subtract -10 from 0.

$$243h^{2}+17h+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 243 for a, 17 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$h=\frac{-17±\sqrt{17^{2}-4\times 243\times 10}}{2\times 243}$$

Square 17.

$$h=\frac{-17±\sqrt{289-4\times 243\times 10}}{2\times 243}$$

Multiply -4 times 243.

$$h=\frac{-17±\sqrt{289-972\times 10}}{2\times 243}$$

Multiply -972 times 10.

$$h=\frac{-17±\sqrt{289-9720}}{2\times 243}$$

Add 289 to -9720.

$$h=\frac{-17±\sqrt{-9431}}{2\times 243}$$

Take the square root of -9431.

$$h=\frac{-17±\sqrt{9431}i}{2\times 243}$$

Multiply 2 times 243.

$$h=\frac{-17±\sqrt{9431}i}{486}$$

Now solve the equation h=\frac{-17±\sqrt{9431}i}{486} when ± is plus. Add -17 to i\sqrt{9431}.

$$h=\frac{-17+\sqrt{9431}i}{486}$$

Now solve the equation h=\frac{-17±\sqrt{9431}i}{486} when ± is minus. Subtract i\sqrt{9431} from -17.

$$h=\frac{-\sqrt{9431}i-17}{486}$$

The equation is now solved.

$$h=\frac{-17+\sqrt{9431}i}{486} h=\frac{-\sqrt{9431}i-17}{486}$$

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.

$$243h^{2}+17h=-10$$

Divide both sides by 243.

$$\frac{243h^{2}+17h}{243}=-\frac{10}{243}$$

Dividing by 243 undoes the multiplication by 243.

$$h^{2}+\frac{17}{243}h=-\frac{10}{243}$$

Divide \frac{17}{243}, the coefficient of the x term, by 2 to get \frac{17}{486}. Then add the square of \frac{17}{486} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$h^{2}+\frac{17}{243}h+\left(\frac{17}{486}\right)^{2}=-\frac{10}{243}+\left(\frac{17}{486}\right)^{2}$$

Square \frac{17}{486} by squaring both the numerator and the denominator of the fraction.

$$h^{2}+\frac{17}{243}h+\frac{289}{236196}=-\frac{10}{243}+\frac{289}{236196}$$

Add -\frac{10}{243} to \frac{289}{236196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$h^{2}+\frac{17}{243}h+\frac{289}{236196}=-\frac{9431}{236196}$$

Factor h^{2}+\frac{17}{243}h+\frac{289}{236196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(h+\frac{17}{486}\right)^{2}=-\frac{9431}{236196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(h+\frac{17}{486}\right)^{2}}=\sqrt{-\frac{9431}{236196}}$$

Simplify.

$$h+\frac{17}{486}=\frac{\sqrt{9431}i}{486} h+\frac{17}{486}=-\frac{\sqrt{9431}i}{486}$$

Subtract \frac{17}{486} from both sides of the equation.

$$h=\frac{-17+\sqrt{9431}i}{486} h=\frac{-\sqrt{9431}i-17}{486}$$