Q:

How to help motivate math when tutoring low level algebra (High school)

I was tutoring a student today and we were doing basic factoring of quadratics and expanding terms like $(x+2)(x+5)$. Now he ended up being able to do this by the end of our 2 and a half hour session, but it was clear that he didn't understand why he should be learning this and didn't see much of a point to it. Now I could try and tell him that factoring is useful later on for simplifying things, partial fractions for integration, etc, but I don't think this is useful as these students don't know about integration and its uses. Is there any application which a high-schooler could understand that would help motivate learning this material (or other applications which just require high school algebra)?
The only thing I could think of was shooting a projectile off a cliff horizontally and trying to figure out how far it goes. 

A:

I was always honest:
You won't use this.  If you don't like math by now, it's probably not going to be useful to you in your career or life, or ever again once you graduate.  If you're like most people, you're going to completely forget this.
That's not why we're doing it.  Ever watch an athlete workout?  Maybe do a bench press?  Regardless of the sport he or she plays, how many times do they have to lie on their back during a game and heave a heavy weight over their chest?  They're not doing bench presses because that's a specific skill they need:  they're doing it to get stronger so they'll be better at what they do need to do.  You're learning this advanced math to make your brain stronger and to be a smarter, more analytical person.  

A:

I think at this level, and for a tutor (not a teacher), the main carrot (really more of a stick) is getting good grades, getting into a good college, etc. Most people really will never use this later, and for most others it's mostly a stepping stone to more advanced topics.
As for how to present it to someone, try the handshake method (every term in one factor "shakes hands" with every term in the other factor) for how to carry it out, and try the rectangle area method for a justification at their level. By "rectangle area method" I mean, when expanding $(x+2)(x+5),$ draw a rectangle with one side of length $x+2$ (the side consists of abutting intervals on the same line, one labeled as length $x$ and the other labeled as length $2),$ and an adjacent side of length $x+5$ (the side consists of abutting intervals on the same line, one labeled as length $x$ and the other labeled as length $5).$
For basic trinomial factoring, look up something called the AC Method. I like this method for weaker students because it makes things a little more systematic for weaker students, and it reinforces the idea of "factoring by grouping", which students are often weak with.
However, if you really want to give applications, maybe psudo-applications like the following will be enough:

Example 1: $(997)(1003)$ equals $(1000-3)(1000 + 3) = (1000)^{2} - 3^2$
Example 2: A $17$ percent increase followed by a $17$ percent decrease is a net decrease since $\left(1 + \frac{17}{100}\right)\left(1 - \frac{17}{100}\right)$ equals $1 - \left(\frac{17}{100}\right)^{2} < 1.$
Example 3: To find the prime factorization of $12^{11} + 12^{12} + 12^{13},$ factor as $12^{11}\left(1 + 12 + 12^{2}\right),$ which equals $4^{11}3^{11}(157) = 2^{22} \cdot 3^{11}\cdot 157.$
Example 4: Determine the exact value of $a^2 - b^2$ if
$$a = 7.0241132301442003123012230341430201$$
$$b = 6.0241132301442003123012230341430201$$
Calculators are allowed for #4.

A:

How many of your students are interested in space exploration?
The quadratic equations became a lot more interesting to me when I learned paths about the sun are ellipses, parabolas or hyperbolas with the sun as a focus. Paths of moons or satellites about a planet are also conic sections with the planet center at a focus.
For me, the visual presentation of quadratics as conic sections also made them more tangible and less remote and abstract.
And if the student becomes interested in quadratics, he might also become interested in factoring them.

