Solve 1/24x^2-23/12x+29=6

$$\frac { 1 } { 24 } x ^ { 2 } - \frac { 23 } { 12 } x + 29 = 6$$

Solve for x (complex solution)

$$x=23+\sqrt{23}i\approx 23+4.795831523i$$

$$x=-\sqrt{23}i+23\approx 23-4.795831523i$$

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.

$$\frac{1}{24}x^{2}-\frac{23}{12}x+29=6$$

Subtract 6 from both sides of the equation.

$$\frac{1}{24}x^{2}-\frac{23}{12}x+29-6=6-6$$

Subtracting 6 from itself leaves 0.

$$\frac{1}{24}x^{2}-\frac{23}{12}x+29-6=0$$

Subtract 6 from 29.

$$\frac{1}{24}x^{2}-\frac{23}{12}x+23=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{24} for a, -\frac{23}{12} for b, and 23 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{23}{12}\right)±\sqrt{\left(-\frac{23}{12}\right)^{2}-4\times \frac{1}{24}\times 23}}{2\times \frac{1}{24}}$$

Square -\frac{23}{12} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{23}{12}\right)±\sqrt{\frac{529}{144}-4\times \frac{1}{24}\times 23}}{2\times \frac{1}{24}}$$

Multiply -4 times \frac{1}{24}.

$$x=\frac{-\left(-\frac{23}{12}\right)±\sqrt{\frac{529}{144}-\frac{1}{6}\times 23}}{2\times \frac{1}{24}}$$

Multiply -\frac{1}{6} times 23.

$$x=\frac{-\left(-\frac{23}{12}\right)±\sqrt{\frac{529}{144}-\frac{23}{6}}}{2\times \frac{1}{24}}$$

Add \frac{529}{144} to -\frac{23}{6} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{23}{12}\right)±\sqrt{-\frac{23}{144}}}{2\times \frac{1}{24}}$$

Take the square root of -\frac{23}{144}.

$$x=\frac{-\left(-\frac{23}{12}\right)±\frac{\sqrt{23}i}{12}}{2\times \frac{1}{24}}$$

The opposite of -\frac{23}{12} is \frac{23}{12}.

$$x=\frac{\frac{23}{12}±\frac{\sqrt{23}i}{12}}{2\times \frac{1}{24}}$$

Multiply 2 times \frac{1}{24}.

$$x=\frac{\frac{23}{12}±\frac{\sqrt{23}i}{12}}{\frac{1}{12}}$$

Now solve the equation x=\frac{\frac{23}{12}±\frac{\sqrt{23}i}{12}}{\frac{1}{12}} when ± is plus. Add \frac{23}{12} to \frac{i\sqrt{23}}{12}.

$$x=\frac{23+\sqrt{23}i}{\frac{1}{12}\times 12}$$

Divide \frac{23+i\sqrt{23}}{12} by \frac{1}{12} by multiplying \frac{23+i\sqrt{23}}{12} by the reciprocal of \frac{1}{12}.

$$x=23+\sqrt{23}i$$

Now solve the equation x=\frac{\frac{23}{12}±\frac{\sqrt{23}i}{12}}{\frac{1}{12}} when ± is minus. Subtract \frac{i\sqrt{23}}{12} from \frac{23}{12}.

$$x=\frac{-\sqrt{23}i+23}{\frac{1}{12}\times 12}$$

Divide \frac{23-i\sqrt{23}}{12} by \frac{1}{12} by multiplying \frac{23-i\sqrt{23}}{12} by the reciprocal of \frac{1}{12}.

$$x=-\sqrt{23}i+23$$

The equation is now solved.

$$x=23+\sqrt{23}i x=-\sqrt{23}i+23$$

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.

$$\frac{1}{24}x^{2}-\frac{23}{12}x+29=6$$

Subtract 29 from both sides of the equation.

$$\frac{1}{24}x^{2}-\frac{23}{12}x+29-29=6-29$$

Subtracting 29 from itself leaves 0.

$$\frac{1}{24}x^{2}-\frac{23}{12}x=6-29$$

Subtract 29 from 6.

$$\frac{1}{24}x^{2}-\frac{23}{12}x=-23$$

Multiply both sides by 24.

$$\frac{\frac{1}{24}x^{2}-\frac{23}{12}x}{\frac{1}{24}}=-\frac{23}{\frac{1}{24}}$$

Dividing by \frac{1}{24} undoes the multiplication by \frac{1}{24}.

$$x^{2}+\left(-\frac{\frac{23}{12}}{\frac{1}{24}}\right)x=-\frac{23}{\frac{1}{24}}$$

Divide -\frac{23}{12} by \frac{1}{24} by multiplying -\frac{23}{12} by the reciprocal of \frac{1}{24}.

$$x^{2}-46x=-\frac{23}{\frac{1}{24}}$$

Divide -23 by \frac{1}{24} by multiplying -23 by the reciprocal of \frac{1}{24}.

$$x^{2}-46x=-552$$

Divide -46, the coefficient of the x term, by 2 to get -23. Then add the square of -23 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-46x+\left(-23\right)^{2}=-552+\left(-23\right)^{2}$$

Square -23.

$$x^{2}-46x+529=-552+529$$

Add -552 to 529.

$$x^{2}-46x+529=-23$$

Factor x^{2}-46x+529. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-23\right)^{2}=-23$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-23\right)^{2}}=\sqrt{-23}$$

Simplify.

$$x-23=\sqrt{23}i x-23=-\sqrt{23}i$$

Add 23 to both sides of the equation.

$$x=23+\sqrt{23}i x=-\sqrt{23}i+23$$