greenplanet2050

Thank you so much!

Actually, I solved it! If anyone needs help? Here a hint: Use difference of squares: \(x^2-y^2=(x+y)(x-y)\) and prime factorization! 

Let's first simpify the quadratic. Moving all terms to the left hand side, we get \(2x^2-2x-10=0 \). We want to find the value of \(a+b+6\). Instead of finding the roots of the quadratic, we can use Vieta's Formula. So the sum of the roots would be \(-\left(-\dfrac{2}{2}\right)\). \(1+6=\boxed{7}\). And now we're done!  Hope this helped!

Let's first rearrange the equation to the standord form. Moving all terms to the right, we get \(19x^2-6x+31=0 \). We apply the quadratic formula and we get the roots are \(3 \pm 25i\)  So \(a\) and \(b\) are \(3\) and \(25\). \(3 \cdot 25=\boxed{75} \). And now, we're done!  Hope this helped!

If you draw a diagram, and calculate, you will get \(\boxed{\dfrac{2}{3}}\)

To find the area of a trapazoid, we need the find the height and the average of the bases of the trapazoid. The height of the trapazoid is \(16\). The average of the bases is \(25\). \(\dfrac{25 \cdot 16}{2}=200 \). And now, we're done!  Hope this helped!

By testing rows of Pascal's triangle, we get row \(3\) satisfies the conditions above! So \(n\) is equal to \(3\). And now we're done!  Hope this helped!

There is only one way to arrange the three yellow squares into the five slots. Now we only have two more slots. Both slots each have \(2\) choices for a color: red or blue. So \(2 \cdot 2 = \boxed{4}\)

That is the number of ways to color the squares. And now, we're done!  Hope this helped!

The formula of finding the volume of a tetrahedron is this: \(\dfrac{a^3}{6 \sqrt2}\) where a is the side length of the tetrahedron. So the volume of this tetrahedron is \(\dfrac{1}{6 \sqrt2}=\dfrac{\sqrt2}{12}\)

And now, we're done!  Hope this helped!

Let's let the final length of the triangle be equal to x. By the Triangle inequality, 

24+49>x

24+x>49

49+x>24.

The first inequality gives us the equation x<73. The second equation gives up x>25. The final equation gives up x>-25. From these equations, we know that 25 <73. So the smallest possible length of x is 26. So the perimeter of the triangle is 26+24+49=99. 

And now we're done!  Hope this helped!