Solve -86d=-92d^2

$$- 86 d = - 92 d ^ { 2 }$$

Solve for d

$$d=\frac{43}{46}\approx 0.934782609$$

$$d=0$$

Steps Using Factoring

Add 92d^{2} to both sides.

$$-86d+92d^{2}=0$$

Factor out d.

$$d\left(-86+92d\right)=0$$

To find equation solutions, solve d=0 and -86+92d=0.

$$d=0 d=\frac{43}{46}$$

Steps Using the Quadratic Formula

Add 92d^{2} to both sides.

$$-86d+92d^{2}=0$$

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.

$$92d^{2}-86d=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 92 for a, -86 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$d=\frac{-\left(-86\right)±\sqrt{\left(-86\right)^{2}}}{2\times 92}$$

Take the square root of \left(-86\right)^{2}.

$$d=\frac{-\left(-86\right)±86}{2\times 92}$$

The opposite of -86 is 86.

$$d=\frac{86±86}{2\times 92}$$

Multiply 2 times 92.

$$d=\frac{86±86}{184}$$

Now solve the equation d=\frac{86±86}{184} when ± is plus. Add 86 to 86.

$$d=\frac{172}{184}$$

Reduce the fraction \frac{172}{184} to lowest terms by extracting and canceling out 4.

$$d=\frac{43}{46}$$

Now solve the equation d=\frac{86±86}{184} when ± is minus. Subtract 86 from 86.

$$d=\frac{0}{184}$$

Divide 0 by 184.

$$d=0$$

The equation is now solved.

$$d=\frac{43}{46} d=0$$

Steps for Completing the Square

Add 92d^{2} to both sides.

$$-86d+92d^{2}=0$$

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.

$$92d^{2}-86d=0$$

Divide both sides by 92.

$$\frac{92d^{2}-86d}{92}=\frac{0}{92}$$

Dividing by 92 undoes the multiplication by 92.

$$d^{2}+\left(-\frac{86}{92}\right)d=\frac{0}{92}$$

Reduce the fraction \frac{-86}{92} to lowest terms by extracting and canceling out 2.

$$d^{2}-\frac{43}{46}d=\frac{0}{92}$$

Divide 0 by 92.

$$d^{2}-\frac{43}{46}d=0$$

Divide -\frac{43}{46}, the coefficient of the x term, by 2 to get -\frac{43}{92}. Then add the square of -\frac{43}{92} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$d^{2}-\frac{43}{46}d+\left(-\frac{43}{92}\right)^{2}=\left(-\frac{43}{92}\right)^{2}$$

Square -\frac{43}{92} by squaring both the numerator and the denominator of the fraction.

$$d^{2}-\frac{43}{46}d+\frac{1849}{8464}=\frac{1849}{8464}$$

Factor d^{2}-\frac{43}{46}d+\frac{1849}{8464}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(d-\frac{43}{92}\right)^{2}=\frac{1849}{8464}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(d-\frac{43}{92}\right)^{2}}=\sqrt{\frac{1849}{8464}}$$

Simplify.

$$d-\frac{43}{92}=\frac{43}{92} d-\frac{43}{92}=-\frac{43}{92}$$

Add \frac{43}{92} to both sides of the equation.

$$d=\frac{43}{46} d=0$$