Solve -20700-153900d=5386d^2

$$-20700-153900d = 5386 { d }^{ 2 }$$

Solve for d

$$d=\frac{15\sqrt{6455347}-38475}{2693}\approx -0.135142083$$

$$d=\frac{-15\sqrt{6455347}-38475}{2693}\approx -28.438938868$$

Steps Using the Quadratic Formula

Subtract 5386d^{2} from both sides.

$$-20700-153900d-5386d^{2}=0$$

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.

$$-5386d^{2}-153900d-20700=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5386 for a, -153900 for b, and -20700 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$d=\frac{-\left(-153900\right)±\sqrt{\left(-153900\right)^{2}-4\left(-5386\right)\left(-20700\right)}}{2\left(-5386\right)}$$

Square -153900.

$$d=\frac{-\left(-153900\right)±\sqrt{23685210000-4\left(-5386\right)\left(-20700\right)}}{2\left(-5386\right)}$$

Multiply -4 times -5386.

$$d=\frac{-\left(-153900\right)±\sqrt{23685210000+21544\left(-20700\right)}}{2\left(-5386\right)}$$

Multiply 21544 times -20700.

$$d=\frac{-\left(-153900\right)±\sqrt{23685210000-445960800}}{2\left(-5386\right)}$$

Add 23685210000 to -445960800.

$$d=\frac{-\left(-153900\right)±\sqrt{23239249200}}{2\left(-5386\right)}$$

Take the square root of 23239249200.

$$d=\frac{-\left(-153900\right)±60\sqrt{6455347}}{2\left(-5386\right)}$$

The opposite of -153900 is 153900.

$$d=\frac{153900±60\sqrt{6455347}}{2\left(-5386\right)}$$

Multiply 2 times -5386.

$$d=\frac{153900±60\sqrt{6455347}}{-10772}$$

Now solve the equation d=\frac{153900±60\sqrt{6455347}}{-10772} when ± is plus. Add 153900 to 60\sqrt{6455347}.

$$d=\frac{60\sqrt{6455347}+153900}{-10772}$$

Divide 153900+60\sqrt{6455347} by -10772.

$$d=\frac{-15\sqrt{6455347}-38475}{2693}$$

Now solve the equation d=\frac{153900±60\sqrt{6455347}}{-10772} when ± is minus. Subtract 60\sqrt{6455347} from 153900.

$$d=\frac{153900-60\sqrt{6455347}}{-10772}$$

Divide 153900-60\sqrt{6455347} by -10772.

$$d=\frac{15\sqrt{6455347}-38475}{2693}$$

The equation is now solved.

$$d=\frac{-15\sqrt{6455347}-38475}{2693} d=\frac{15\sqrt{6455347}-38475}{2693}$$

Steps for Completing the Square

Subtract 5386d^{2} from both sides.

$$-20700-153900d-5386d^{2}=0$$

Add 20700 to both sides. Anything plus zero gives itself.

$$-153900d-5386d^{2}=20700$$

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.

$$-5386d^{2}-153900d=20700$$

Divide both sides by -5386.

$$\frac{-5386d^{2}-153900d}{-5386}=\frac{20700}{-5386}$$

Dividing by -5386 undoes the multiplication by -5386.

$$d^{2}+\left(-\frac{153900}{-5386}\right)d=\frac{20700}{-5386}$$

Reduce the fraction \frac{-153900}{-5386} to lowest terms by extracting and canceling out 2.

$$d^{2}+\frac{76950}{2693}d=\frac{20700}{-5386}$$

Reduce the fraction \frac{20700}{-5386} to lowest terms by extracting and canceling out 2.

$$d^{2}+\frac{76950}{2693}d=-\frac{10350}{2693}$$

Divide \frac{76950}{2693}, the coefficient of the x term, by 2 to get \frac{38475}{2693}. Then add the square of \frac{38475}{2693} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$d^{2}+\frac{76950}{2693}d+\left(\frac{38475}{2693}\right)^{2}=-\frac{10350}{2693}+\left(\frac{38475}{2693}\right)^{2}$$

Square \frac{38475}{2693} by squaring both the numerator and the denominator of the fraction.

$$d^{2}+\frac{76950}{2693}d+\frac{1480325625}{7252249}=-\frac{10350}{2693}+\frac{1480325625}{7252249}$$

Add -\frac{10350}{2693} to \frac{1480325625}{7252249} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$d^{2}+\frac{76950}{2693}d+\frac{1480325625}{7252249}=\frac{1452453075}{7252249}$$

Factor d^{2}+\frac{76950}{2693}d+\frac{1480325625}{7252249}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(d+\frac{38475}{2693}\right)^{2}=\frac{1452453075}{7252249}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(d+\frac{38475}{2693}\right)^{2}}=\sqrt{\frac{1452453075}{7252249}}$$

Simplify.

$$d+\frac{38475}{2693}=\frac{15\sqrt{6455347}}{2693} d+\frac{38475}{2693}=-\frac{15\sqrt{6455347}}{2693}$$

Subtract \frac{38475}{2693} from both sides of the equation.

$$d=\frac{15\sqrt{6455347}-38475}{2693} d=\frac{-15\sqrt{6455347}-38475}{2693}$$