Solve 24.3h^2+17h=-10

$$24.3 h ^ { 2 } + 17 h = - 10$$

Solve for h

$$h=\frac{-85+i\times 5\sqrt{683}}{243}\approx -0.349794239+0.537742154i$$

$$h=\frac{-i\times 5\sqrt{683}-85}{243}\approx -0.349794239-0.537742154i$$

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.

$$24.3h^{2}+17h=-10$$

Add 10 to both sides of the equation.

$$24.3h^{2}+17h-\left(-10\right)=-10-\left(-10\right)$$

Subtracting -10 from itself leaves 0.

$$24.3h^{2}+17h-\left(-10\right)=0$$

Subtract -10 from 0.

$$24.3h^{2}+17h+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 24.3 for a, 17 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$h=\frac{-17±\sqrt{17^{2}-4\times 24.3\times 10}}{2\times 24.3}$$

Square 17.

$$h=\frac{-17±\sqrt{289-4\times 24.3\times 10}}{2\times 24.3}$$

Multiply -4 times 24.3.

$$h=\frac{-17±\sqrt{289-97.2\times 10}}{2\times 24.3}$$

Multiply -97.2 times 10.

$$h=\frac{-17±\sqrt{289-972}}{2\times 24.3}$$

Add 289 to -972.

$$h=\frac{-17±\sqrt{-683}}{2\times 24.3}$$

Take the square root of -683.

$$h=\frac{-17±\sqrt{683}i}{2\times 24.3}$$

Multiply 2 times 24.3.

$$h=\frac{-17±\sqrt{683}i}{48.6}$$

Now solve the equation h=\frac{-17±\sqrt{683}i}{48.6} when ± is plus. Add -17 to i\sqrt{683}.

$$h=\frac{-17+\sqrt{683}i}{48.6}$$

Divide -17+i\sqrt{683} by 48.6 by multiplying -17+i\sqrt{683} by the reciprocal of 48.6.

$$h=\frac{-85+5\sqrt{683}i}{243}$$

Now solve the equation h=\frac{-17±\sqrt{683}i}{48.6} when ± is minus. Subtract i\sqrt{683} from -17.

$$h=\frac{-\sqrt{683}i-17}{48.6}$$

Divide -17-i\sqrt{683} by 48.6 by multiplying -17-i\sqrt{683} by the reciprocal of 48.6.

$$h=\frac{-5\sqrt{683}i-85}{243}$$

The equation is now solved.

$$h=\frac{-85+5\sqrt{683}i}{243} h=\frac{-5\sqrt{683}i-85}{243}$$

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.

$$24.3h^{2}+17h=-10$$

Divide both sides of the equation by 24.3, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{24.3h^{2}+17h}{24.3}=-\frac{10}{24.3}$$

Dividing by 24.3 undoes the multiplication by 24.3.

$$h^{2}+\frac{17}{24.3}h=-\frac{10}{24.3}$$

Divide 17 by 24.3 by multiplying 17 by the reciprocal of 24.3.

$$h^{2}+\frac{170}{243}h=-\frac{10}{24.3}$$

Divide -10 by 24.3 by multiplying -10 by the reciprocal of 24.3.

$$h^{2}+\frac{170}{243}h=-\frac{100}{243}$$

Divide \frac{170}{243}, the coefficient of the x term, by 2 to get \frac{85}{243}. Then add the square of \frac{85}{243} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$h^{2}+\frac{170}{243}h+\frac{85}{243}^{2}=-\frac{100}{243}+\frac{85}{243}^{2}$$

Square \frac{85}{243} by squaring both the numerator and the denominator of the fraction.

$$h^{2}+\frac{170}{243}h+\frac{7225}{59049}=-\frac{100}{243}+\frac{7225}{59049}$$

Add -\frac{100}{243} to \frac{7225}{59049} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$h^{2}+\frac{170}{243}h+\frac{7225}{59049}=-\frac{17075}{59049}$$

Factor h^{2}+\frac{170}{243}h+\frac{7225}{59049}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(h+\frac{85}{243}\right)^{2}=-\frac{17075}{59049}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(h+\frac{85}{243}\right)^{2}}=\sqrt{-\frac{17075}{59049}}$$

Simplify.

$$h+\frac{85}{243}=\frac{5\sqrt{683}i}{243} h+\frac{85}{243}=-\frac{5\sqrt{683}i}{243}$$

Subtract \frac{85}{243} from both sides of the equation.

$$h=\frac{-85+5\sqrt{683}i}{243} h=\frac{-5\sqrt{683}i-85}{243}$$