Community Activity:Henry Hudson Math

Since I have nothing to do for the next 2 weeks, I will be starting a free math service for those who are required to do online schooling.

im new, how does this work?
Just post your math problems on this page, preferably an image or PDF. Please do not write out the equations on this page using text, it's confusing.

I will give answers, if you want solutions please request them.

Unless I am really busy (unlikely), I can get problems back to you within a few hours of the request.

Why am I doing this?
Like many, my school has been canceled for the next two weeks, and I don't get online school so I can help others I guess lol. Also, I'm a pretty big math nerd and I'm can do mental math really fast.

Ok have fun.

Questions
ComicsCreatorz (This is for fun.)

Find the constant k so that : -x2 - (k + 7)x - 8 = -(x - 2)(x - 4)

I assume

$$x2=2x$$ .

Multiplying both sides by -1, we have

$$2x+(k+7)x+8=(x-2)(x-4) $$ . Expanding the right side,

$$2x+(k+7)x+8=x^2-6x+8.$$ Moving all terms to the RHS, we have

$$0=x^2-(k+15)x+8 $$ In order for real solutions, the discriminant must be greater than equal to

$$0$$ . Knowing that, we can set

$$(k+15)^2-32 \geq 0. $$ Adding both sides by

$$32$$, we have

$$(k+15)^2 \geq 32. $$ This is valid for all

$$k \geq 4 \sqrt{2}-15.$$ Therefore, values of k in the interval

$$ [4\sqrt{2}-15, \infty) $$ are valid.

There is no specific answer because we do not know x.

Spyroclub1 (This is for my actual work)

What two numbers add up to -1 and multiply to -15?

We have $$m+n=-1, mn=-15$$ By Vieta's, we have the quadratic $$x^2+x-15=0$$ By the Quadratic Formula, our roots are $$\frac{-1 \pm \sqrt{61}}{2}$$. Therefore our answer is $$\boxed{\frac{-1 + \sqrt{61}}{2}} $$ and $$ \boxed{\frac{-1 - \sqrt{61}}{2}}$$