Solve 3-13d=3-8d+d^2

$$3 - 13 d = 3 - 8 d + d ^ { 2 }$$

Solve for d

$$d=-5$$

$$d=0$$

Steps Using Factoring

Subtract 3 from both sides.

$$3-13d-3=-8d+d^{2}$$

Subtract 3 from 3 to get 0.

$$-13d=-8d+d^{2}$$

Add 8d to both sides.

$$-13d+8d=d^{2}$$

Combine -13d and 8d to get -5d.

$$-5d=d^{2}$$

Subtract d^{2} from both sides.

$$-5d-d^{2}=0$$

Factor out d.

$$d\left(-5-d\right)=0$$

To find equation solutions, solve d=0 and -5-d=0.

$$d=0 d=-5$$

Steps Using the Quadratic Formula

Subtract 3 from both sides.

$$3-13d-3=-8d+d^{2}$$

Subtract 3 from 3 to get 0.

$$-13d=-8d+d^{2}$$

Add 8d to both sides.

$$-13d+8d=d^{2}$$

Combine -13d and 8d to get -5d.

$$-5d=d^{2}$$

Subtract d^{2} from both sides.

$$-5d-d^{2}=0$$

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.

$$-d^{2}-5d=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -5 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$d=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}}}{2\left(-1\right)}$$

Take the square root of \left(-5\right)^{2}.

$$d=\frac{-\left(-5\right)±5}{2\left(-1\right)}$$

The opposite of -5 is 5.

$$d=\frac{5±5}{2\left(-1\right)}$$

Multiply 2 times -1.

$$d=\frac{5±5}{-2}$$

Now solve the equation d=\frac{5±5}{-2} when ± is plus. Add 5 to 5.

$$d=\frac{10}{-2}$$

Divide 10 by -2.

$$d=-5$$

Now solve the equation d=\frac{5±5}{-2} when ± is minus. Subtract 5 from 5.

$$d=\frac{0}{-2}$$

Divide 0 by -2.

$$d=0$$

The equation is now solved.

$$d=-5 d=0$$

Steps for Completing the Square

Add 8d to both sides.

$$3-13d+8d=3+d^{2}$$

Combine -13d and 8d to get -5d.

$$3-5d=3+d^{2}$$

Subtract d^{2} from both sides.

$$3-5d-d^{2}=3$$

Subtract 3 from both sides.

$$-5d-d^{2}=3-3$$

Subtract 3 from 3 to get 0.

$$-5d-d^{2}=0$$

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.

$$-d^{2}-5d=0$$

Divide both sides by -1.

$$\frac{-d^{2}-5d}{-1}=\frac{0}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$d^{2}+\left(-\frac{5}{-1}\right)d=\frac{0}{-1}$$

Divide -5 by -1.

$$d^{2}+5d=\frac{0}{-1}$$

Divide 0 by -1.

$$d^{2}+5d=0$$

Divide 5, the coefficient of the x term, by 2 to get \frac{5}{2}. Then add the square of \frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$d^{2}+5d+\left(\frac{5}{2}\right)^{2}=\left(\frac{5}{2}\right)^{2}$$

Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$d^{2}+5d+\frac{25}{4}=\frac{25}{4}$$

Factor d^{2}+5d+\frac{25}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(d+\frac{5}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(d+\frac{5}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$d+\frac{5}{2}=\frac{5}{2} d+\frac{5}{2}=-\frac{5}{2}$$

Subtract \frac{5}{2} from both sides of the equation.

$$d=0 d=-5$$