Solve N(t)=-t^2-10t+1200

$$N ( t ) = - t ^ { 2 } - 10 t + 1200$$

Factor

$$\left(30-t\right)\left(t+40\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -t^{2}+at+bt+1200. To find a and b, set up a system to be solved.

$$a+b=-10 ab=-1200=-1200$$

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 -1200.

$$1,-1200 2,-600 3,-400 4,-300 5,-240 6,-200 8,-150 10,-120 12,-100 15,-80 16,-75 20,-60 24,-50 25,-48 30,-40$$

Calculate the sum for each pair.

$$1-1200=-1199 2-600=-598 3-400=-397 4-300=-296 5-240=-235 6-200=-194 8-150=-142 10-120=-110 12-100=-88 15-80=-65 16-75=-59 20-60=-40 24-50=-26 25-48=-23 30-40=-10$$

The solution is the pair that gives sum -10.

$$a=30 b=-40$$

Rewrite -t^{2}-10t+1200 as \left(-t^{2}+30t\right)+\left(-40t+1200\right).

$$\left(-t^{2}+30t\right)+\left(-40t+1200\right)$$

Factor out t in the first and 40 in the second group.

$$t\left(-t+30\right)+40\left(-t+30\right)$$

Factor out common term -t+30 by using distributive property.

$$\left(-t+30\right)\left(t+40\right)$$

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.

$$-t^{2}-10t+1200=0$$

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.

$$t=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-1\right)\times 1200}}{2\left(-1\right)}$$

Square -10.

$$t=\frac{-\left(-10\right)±\sqrt{100-4\left(-1\right)\times 1200}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$t=\frac{-\left(-10\right)±\sqrt{100+4\times 1200}}{2\left(-1\right)}$$

Multiply 4 times 1200.

$$t=\frac{-\left(-10\right)±\sqrt{100+4800}}{2\left(-1\right)}$$

Add 100 to 4800.

$$t=\frac{-\left(-10\right)±\sqrt{4900}}{2\left(-1\right)}$$

Take the square root of 4900.

$$t=\frac{-\left(-10\right)±70}{2\left(-1\right)}$$

The opposite of -10 is 10.

$$t=\frac{10±70}{2\left(-1\right)}$$

Multiply 2 times -1.

$$t=\frac{10±70}{-2}$$

Now solve the equation t=\frac{10±70}{-2} when ± is plus. Add 10 to 70.

$$t=\frac{80}{-2}$$

Divide 80 by -2.

$$t=-40$$

Now solve the equation t=\frac{10±70}{-2} when ± is minus. Subtract 70 from 10.

$$t=-\frac{60}{-2}$$

Divide -60 by -2.

$$t=30$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -40 for x_{1} and 30 for x_{2}.

$$-t^{2}-10t+1200=-\left(t-\left(-40\right)\right)\left(t-30\right)$$

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

$$-t^{2}-10t+1200=-\left(t+40\right)\left(t-30\right)$$

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.

$$x ^ 2 +10x -1200 = 0$$

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

$$r + s = -10 rs = -1200$$

Two numbers r and s sum up to -10 exactly when the average of the two numbers is \frac{1}{2}*-10 = -5. 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>

$$r = -5 - u s = -5 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -1200

$$(-5 - u) (-5 + u) = -1200$$

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

$$25 - u^2 = -1200$$

Simplify the expression by subtracting 25 on both sides

$$-u^2 = -1200-25 = -1225$$

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

$$u^2 = 1225 u = \pm\sqrt{1225} = \pm 35$$

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

$$r =-5 - 35 = -40 s = -5 + 35 = 30$$

Evaluate

$$\left(30-t\right)\left(t+40\right)$$