Solve -16t^2+100t+600=

$$- 16 t ^ { 2 } + 100 t + 600 =$$

Factor

$$4\left(10-t\right)\left(4t+15\right)$$

Solution Steps

Factor out 4.

$$4\left(-4t^{2}+25t+150\right)$$

Consider -4t^{2}+25t+150. Factor the expression by grouping. First, the expression needs to be rewritten as -4t^{2}+at+bt+150. To find a and b, set up a system to be solved.

$$a+b=25 ab=-4\times 150=-600$$

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 -600.

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

Calculate the sum for each pair.

$$-1+600=599 -2+300=298 -3+200=197 -4+150=146 -5+120=115 -6+100=94 -8+75=67 -10+60=50 -12+50=38 -15+40=25 -20+30=10 -24+25=1$$

The solution is the pair that gives sum 25.

$$a=40 b=-15$$

Rewrite -4t^{2}+25t+150 as \left(-4t^{2}+40t\right)+\left(-15t+150\right).

$$\left(-4t^{2}+40t\right)+\left(-15t+150\right)$$

Factor out 4t in the first and 15 in the second group.

$$4t\left(-t+10\right)+15\left(-t+10\right)$$

Factor out common term -t+10 by using distributive property.

$$\left(-t+10\right)\left(4t+15\right)$$

Rewrite the complete factored expression.

$$4\left(-t+10\right)\left(4t+15\right)$$

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.

$$-16t^{2}+100t+600=0$$

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.

$$t=\frac{-100±\sqrt{100^{2}-4\left(-16\right)\times 600}}{2\left(-16\right)}$$

Square 100.

$$t=\frac{-100±\sqrt{10000-4\left(-16\right)\times 600}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$t=\frac{-100±\sqrt{10000+64\times 600}}{2\left(-16\right)}$$

Multiply 64 times 600.

$$t=\frac{-100±\sqrt{10000+38400}}{2\left(-16\right)}$$

Add 10000 to 38400.

$$t=\frac{-100±\sqrt{48400}}{2\left(-16\right)}$$

Take the square root of 48400.

$$t=\frac{-100±220}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{-100±220}{-32}$$

Now solve the equation t=\frac{-100±220}{-32} when ± is plus. Add -100 to 220.

$$t=\frac{120}{-32}$$

Reduce the fraction \frac{120}{-32} to lowest terms by extracting and canceling out 8.

$$t=-\frac{15}{4}$$

Now solve the equation t=\frac{-100±220}{-32} when ± is minus. Subtract 220 from -100.

$$t=-\frac{320}{-32}$$

Divide -320 by -32.

$$t=10$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{15}{4} for x_{1} and 10 for x_{2}.

$$-16t^{2}+100t+600=-16\left(t-\left(-\frac{15}{4}\right)\right)\left(t-10\right)$$

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

$$-16t^{2}+100t+600=-16\left(t+\frac{15}{4}\right)\left(t-10\right)$$

Add \frac{15}{4} to t by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-16t^{2}+100t+600=-16\times \frac{-4t-15}{-4}\left(t-10\right)$$

Cancel out 4, the greatest common factor in -16 and 4.

$$-16t^{2}+100t+600=4\left(-4t-15\right)\left(t-10\right)$$

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.

$$x ^ 2 -\frac{25}{4}x -\frac{75}{2} = 0$$

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

$$r + s = \frac{25}{4} rs = -\frac{75}{2}$$

Two numbers r and s sum up to \frac{25}{4} exactly when the average of the two numbers is \frac{1}{2}*\frac{25}{4} = \frac{25}{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-gzdabgg4ehffg0hf.b01.azurefd.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{25}{8} - u s = \frac{25}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{75}{2}

$$(\frac{25}{8} - u) (\frac{25}{8} + u) = -\frac{75}{2}$$

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

$$\frac{625}{64} - u^2 = -\frac{75}{2}$$

Simplify the expression by subtracting \frac{625}{64} on both sides

$$-u^2 = -\frac{75}{2}-\frac{625}{64} = -\frac{3025}{64}$$

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

$$u^2 = \frac{3025}{64} u = \pm\sqrt{\frac{3025}{64}} = \pm \frac{55}{8}$$

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

$$r =\frac{25}{8} - \frac{55}{8} = -3.750 s = \frac{25}{8} + \frac{55}{8} = 10$$

Evaluate

$$600+100t-16t^{2}$$