Solve D(t)=-16t^2-12t+990

$$D ( t ) = - 16 t ^ { 2 } - 12 t + 990$$

Factor

$$2\left(-4t-33\right)\left(2t-15\right)$$

Solution Steps

Factor out 2.

$$2\left(-8t^{2}-6t+495\right)$$

Consider -8t^{2}-6t+495. Factor the expression by grouping. First, the expression needs to be rewritten as -8t^{2}+at+bt+495. To find a and b, set up a system to be solved.

$$a+b=-6 ab=-8\times 495=-3960$$

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 -3960.

$$1,-3960 2,-1980 3,-1320 4,-990 5,-792 6,-660 8,-495 9,-440 10,-396 11,-360 12,-330 15,-264 18,-220 20,-198 22,-180 24,-165 30,-132 33,-120 36,-110 40,-99 44,-90 45,-88 55,-72 60,-66$$

Calculate the sum for each pair.

$$1-3960=-3959 2-1980=-1978 3-1320=-1317 4-990=-986 5-792=-787 6-660=-654 8-495=-487 9-440=-431 10-396=-386 11-360=-349 12-330=-318 15-264=-249 18-220=-202 20-198=-178 22-180=-158 24-165=-141 30-132=-102 33-120=-87 36-110=-74 40-99=-59 44-90=-46 45-88=-43 55-72=-17 60-66=-6$$

The solution is the pair that gives sum -6.

$$a=60 b=-66$$

Rewrite -8t^{2}-6t+495 as \left(-8t^{2}+60t\right)+\left(-66t+495\right).

$$\left(-8t^{2}+60t\right)+\left(-66t+495\right)$$

Factor out -4t in the first and -33 in the second group.

$$-4t\left(2t-15\right)-33\left(2t-15\right)$$

Factor out common term 2t-15 by using distributive property.

$$\left(2t-15\right)\left(-4t-33\right)$$

Rewrite the complete factored expression.

$$2\left(2t-15\right)\left(-4t-33\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-16t^{2}-12t+990=0$$

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.

$$t=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-16\right)\times 990}}{2\left(-16\right)}$$

Square -12.

$$t=\frac{-\left(-12\right)±\sqrt{144-4\left(-16\right)\times 990}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$t=\frac{-\left(-12\right)±\sqrt{144+64\times 990}}{2\left(-16\right)}$$

Multiply 64 times 990.

$$t=\frac{-\left(-12\right)±\sqrt{144+63360}}{2\left(-16\right)}$$

Add 144 to 63360.

$$t=\frac{-\left(-12\right)±\sqrt{63504}}{2\left(-16\right)}$$

Take the square root of 63504.

$$t=\frac{-\left(-12\right)±252}{2\left(-16\right)}$$

The opposite of -12 is 12.

$$t=\frac{12±252}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{12±252}{-32}$$

Now solve the equation t=\frac{12±252}{-32} when ± is plus. Add 12 to 252.

$$t=\frac{264}{-32}$$

Reduce the fraction \frac{264}{-32} to lowest terms by extracting and canceling out 8.

$$t=-\frac{33}{4}$$

Now solve the equation t=\frac{12±252}{-32} when ± is minus. Subtract 252 from 12.

$$t=-\frac{240}{-32}$$

Reduce the fraction \frac{-240}{-32} to lowest terms by extracting and canceling out 16.

$$t=\frac{15}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{33}{4} for x_{1} and \frac{15}{2} for x_{2}.

$$-16t^{2}-12t+990=-16\left(t-\left(-\frac{33}{4}\right)\right)\left(t-\frac{15}{2}\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$-16t^{2}-12t+990=-16\left(t+\frac{33}{4}\right)\left(t-\frac{15}{2}\right)$$

Add \frac{33}{4} to t by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-16t^{2}-12t+990=-16\times \frac{-4t-33}{-4}\left(t-\frac{15}{2}\right)$$

Subtract \frac{15}{2} from t by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-16t^{2}-12t+990=-16\times \frac{-4t-33}{-4}\times \frac{-2t+15}{-2}$$

Multiply \frac{-4t-33}{-4} times \frac{-2t+15}{-2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$-16t^{2}-12t+990=-16\times \frac{\left(-4t-33\right)\left(-2t+15\right)}{-4\left(-2\right)}$$

Multiply -4 times -2.

$$-16t^{2}-12t+990=-16\times \frac{\left(-4t-33\right)\left(-2t+15\right)}{8}$$

Cancel out 8, the greatest common factor in -16 and 8.

$$-16t^{2}-12t+990=-2\left(-4t-33\right)\left(-2t+15\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.

$$x ^ 2 +\frac{3}{4}x -\frac{495}{8} = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = -\frac{3}{4} rs = -\frac{495}{8}$$

Two numbers r and s sum up to -\frac{3}{4} exactly when the average of the two numbers is \frac{1}{2}*-\frac{3}{4} = -\frac{3}{8}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -\frac{3}{8} - u s = -\frac{3}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{495}{8}

$$(-\frac{3}{8} - u) (-\frac{3}{8} + u) = -\frac{495}{8}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{9}{64} - u^2 = -\frac{495}{8}$$

Simplify the expression by subtracting \frac{9}{64} on both sides

$$-u^2 = -\frac{495}{8}-\frac{9}{64} = -\frac{3969}{64}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{3969}{64} u = \pm\sqrt{\frac{3969}{64}} = \pm \frac{63}{8}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =-\frac{3}{8} - \frac{63}{8} = -8.250 s = -\frac{3}{8} + \frac{63}{8} = 7.500$$

Evaluate

$$990-12t-16t^{2}$$