Solve -t^2+38t+4400

$$- t ^ { 2 } + 38 t + 4400$$

Factor

$$-\left(t-88\right)\left(t+50\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -t^{2}+at+bt+4400. To find a and b, set up a system to be solved.

$$a+b=38 ab=-4400=-4400$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -4400.

$$-1,4400 -2,2200 -4,1100 -5,880 -8,550 -10,440 -11,400 -16,275 -20,220 -22,200 -25,176 -40,110 -44,100 -50,88 -55,80$$

Calculate the sum for each pair.

$$-1+4400=4399 -2+2200=2198 -4+1100=1096 -5+880=875 -8+550=542 -10+440=430 -11+400=389 -16+275=259 -20+220=200 -22+200=178 -25+176=151 -40+110=70 -44+100=56 -50+88=38 -55+80=25$$

The solution is the pair that gives sum 38.

$$a=88 b=-50$$

Rewrite -t^{2}+38t+4400 as \left(-t^{2}+88t\right)+\left(-50t+4400\right).

$$\left(-t^{2}+88t\right)+\left(-50t+4400\right)$$

Factor out -t in the first and -50 in the second group.

$$-t\left(t-88\right)-50\left(t-88\right)$$

Factor out common term t-88 by using distributive property.

$$\left(t-88\right)\left(-t-50\right)$$

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.

$$-t^{2}+38t+4400=0$$

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.

$$t=\frac{-38±\sqrt{38^{2}-4\left(-1\right)\times 4400}}{2\left(-1\right)}$$

Square 38.

$$t=\frac{-38±\sqrt{1444-4\left(-1\right)\times 4400}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$t=\frac{-38±\sqrt{1444+4\times 4400}}{2\left(-1\right)}$$

Multiply 4 times 4400.

$$t=\frac{-38±\sqrt{1444+17600}}{2\left(-1\right)}$$

Add 1444 to 17600.

$$t=\frac{-38±\sqrt{19044}}{2\left(-1\right)}$$

Take the square root of 19044.

$$t=\frac{-38±138}{2\left(-1\right)}$$

Multiply 2 times -1.

$$t=\frac{-38±138}{-2}$$

Now solve the equation t=\frac{-38±138}{-2} when ± is plus. Add -38 to 138.

$$t=\frac{100}{-2}$$

Divide 100 by -2.

$$t=-50$$

Now solve the equation t=\frac{-38±138}{-2} when ± is minus. Subtract 138 from -38.

$$t=-\frac{176}{-2}$$

Divide -176 by -2.

$$t=88$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -50 for x_{1} and 88 for x_{2}.

$$-t^{2}+38t+4400=-\left(t-\left(-50\right)\right)\left(t-88\right)$$

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

$$-t^{2}+38t+4400=-\left(t+50\right)\left(t-88\right)$$

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.

$$x ^ 2 -38x -4400 = 0$$

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

$$r + s = 38 rs = -4400$$

Two numbers r and s sum up to 38 exactly when the average of the two numbers is \frac{1}{2}*38 = 19. 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>

$$r = 19 - u s = 19 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -4400

$$(19 - u) (19 + u) = -4400$$

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

$$361 - u^2 = -4400$$

Simplify the expression by subtracting 361 on both sides

$$-u^2 = -4400-361 = -4761$$

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

$$u^2 = 4761 u = \pm\sqrt{4761} = \pm 69$$

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

$$r =19 - 69 = -50 s = 19 + 69 = 88$$

Evaluate

$$-\left(t-88\right)\left(t+50\right)$$