Solve 44d^2-36d-45=0

$$44 { d }^{ 2 } -36d-45=0$$

Solve for d

$$d=-\frac{15}{22}\approx -0.681818182$$

$$d = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 44d^{2}+ad+bd-45. To find a and b, set up a system to be solved.

$$a+b=-36 ab=44\left(-45\right)=-1980$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -1980.

$$1,-1980 2,-990 3,-660 4,-495 5,-396 6,-330 9,-220 10,-198 11,-180 12,-165 15,-132 18,-110 20,-99 22,-90 30,-66 33,-60 36,-55 44,-45$$

Calculate the sum for each pair.

$$1-1980=-1979 2-990=-988 3-660=-657 4-495=-491 5-396=-391 6-330=-324 9-220=-211 10-198=-188 11-180=-169 12-165=-153 15-132=-117 18-110=-92 20-99=-79 22-90=-68 30-66=-36 33-60=-27 36-55=-19 44-45=-1$$

The solution is the pair that gives sum -36.

$$a=-66 b=30$$

Rewrite 44d^{2}-36d-45 as \left(44d^{2}-66d\right)+\left(30d-45\right).

$$\left(44d^{2}-66d\right)+\left(30d-45\right)$$

Factor out 22d in the first and 15 in the second group.

$$22d\left(2d-3\right)+15\left(2d-3\right)$$

Factor out common term 2d-3 by using distributive property.

$$\left(2d-3\right)\left(22d+15\right)$$

To find equation solutions, solve 2d-3=0 and 22d+15=0.

$$d=\frac{3}{2} d=-\frac{15}{22}$$

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.

$$44d^{2}-36d-45=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 44 for a, -36 for b, and -45 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$d=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 44\left(-45\right)}}{2\times 44}$$

Square -36.

$$d=\frac{-\left(-36\right)±\sqrt{1296-4\times 44\left(-45\right)}}{2\times 44}$$

Multiply -4 times 44.

$$d=\frac{-\left(-36\right)±\sqrt{1296-176\left(-45\right)}}{2\times 44}$$

Multiply -176 times -45.

$$d=\frac{-\left(-36\right)±\sqrt{1296+7920}}{2\times 44}$$

Add 1296 to 7920.

$$d=\frac{-\left(-36\right)±\sqrt{9216}}{2\times 44}$$

Take the square root of 9216.

$$d=\frac{-\left(-36\right)±96}{2\times 44}$$

The opposite of -36 is 36.

$$d=\frac{36±96}{2\times 44}$$

Multiply 2 times 44.

$$d=\frac{36±96}{88}$$

Now solve the equation d=\frac{36±96}{88} when ± is plus. Add 36 to 96.

$$d=\frac{132}{88}$$

Reduce the fraction \frac{132}{88} to lowest terms by extracting and canceling out 44.

$$d=\frac{3}{2}$$

Now solve the equation d=\frac{36±96}{88} when ± is minus. Subtract 96 from 36.

$$d=-\frac{60}{88}$$

Reduce the fraction \frac{-60}{88} to lowest terms by extracting and canceling out 4.

$$d=-\frac{15}{22}$$

The equation is now solved.

$$d=\frac{3}{2} d=-\frac{15}{22}$$

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.

$$44d^{2}-36d-45=0$$

Add 45 to both sides of the equation.

$$44d^{2}-36d-45-\left(-45\right)=-\left(-45\right)$$

Subtracting -45 from itself leaves 0.

$$44d^{2}-36d=-\left(-45\right)$$

Subtract -45 from 0.

$$44d^{2}-36d=45$$

Divide both sides by 44.

$$\frac{44d^{2}-36d}{44}=\frac{45}{44}$$

Dividing by 44 undoes the multiplication by 44.

$$d^{2}+\left(-\frac{36}{44}\right)d=\frac{45}{44}$$

Reduce the fraction \frac{-36}{44} to lowest terms by extracting and canceling out 4.

$$d^{2}-\frac{9}{11}d=\frac{45}{44}$$

Divide -\frac{9}{11}, the coefficient of the x term, by 2 to get -\frac{9}{22}. Then add the square of -\frac{9}{22} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$d^{2}-\frac{9}{11}d+\left(-\frac{9}{22}\right)^{2}=\frac{45}{44}+\left(-\frac{9}{22}\right)^{2}$$

Square -\frac{9}{22} by squaring both the numerator and the denominator of the fraction.

$$d^{2}-\frac{9}{11}d+\frac{81}{484}=\frac{45}{44}+\frac{81}{484}$$

Add \frac{45}{44} to \frac{81}{484} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$d^{2}-\frac{9}{11}d+\frac{81}{484}=\frac{144}{121}$$

Factor d^{2}-\frac{9}{11}d+\frac{81}{484}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(d-\frac{9}{22}\right)^{2}=\frac{144}{121}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(d-\frac{9}{22}\right)^{2}}=\sqrt{\frac{144}{121}}$$

Simplify.

$$d-\frac{9}{22}=\frac{12}{11} d-\frac{9}{22}=-\frac{12}{11}$$

Add \frac{9}{22} to both sides of the equation.

$$d=\frac{3}{2} d=-\frac{15}{22}$$