1. Quadratics, Inequalities and Square Forms

Just a few questions I need help on so I can do the rest of my homework.

I have a question which I dont understand very well, the question is 'Solve the following inequalities by sketching by hand the curves of the functions involved' I dont understand exactly what to do with say an equation like p^2-5p+4<0 and 3x^2+5x-2<0

Square Form
How would I do:
2x^2+4x+6

-x^2-2x+5

-x^2+3x+10

2. Originally Posted by Yppolitia
Just a few questions I need help on so I can do the rest of my homework.

I have a question which I dont understand very well, the question is 'Solve the following inequalities by sketching by hand the curves of the functions involved' I dont understand exactly what to do with say an equation like p^2-5p+4<0 and 3x^2+5x-2<0

Square Form
How would I do:
2x^2+4x+6

-x^2-2x+5

-x^2+3x+10
Well, for starters you haven't given us an inequality to solve...

Let's assume we have $\displaystyle 2x^2 + 4x - 6 > 0$.

The first way to do this is to sketch the graph of $\displaystyle y = 2x^2 + 4x - 6$ and then see where y > 0. By looking at the graph below (hopefully it will load) you can see what the solution set will be: $\displaystyle (-\infty, -3) \cup (1, \infty)$.

The other way:
We wish to find the critical points of the function $\displaystyle 2x^2 + 4x - 6$. These are the points where:
1) The function is 0.
2) The denominator is 0.
3) The function under the radical is 0.

For this function, the only critical points are where the function is 0. So solve:
$\displaystyle 2x^2 + 4x - 6 = 0$ Divide both sides by 2:

$\displaystyle x^2 + 2x - 3 = 0$

$\displaystyle (x + 3)(x - 1) = 0$

So x = -3 and x = 1 are the critical points.

Now we want to break the real line into intervals and test the inequality on each interval:
$\displaystyle (-\infty, -3)$: $\displaystyle 2x^2 + 4x - 6 > 0$ (Check!)
$\displaystyle (-3, 1)$: $\displaystyle 2x^2 + 4x - 6 < 0$ (Nope!)
$\displaystyle (1, \infty)$: $\displaystyle 2x^2 + 4x - 6 > 0$ (Check!)

So we see the solution set is: $\displaystyle (-\infty, -3) \cup (1, \infty)$ as we saw from the graph.

-Dan

3. So for the answer of the quesiotn, please write i completed square form '2x^2+4x+6' what do I put?

4. Originally Posted by Yppolitia
So for the answer of the quesiotn, please write i completed square form '2x^2+4x+6' what do I put?
Alright, perhaps I didn't understand what you meant. I had thought your question was about how to solve quadratic inequalities. Can you give me an example of what you mean by "write i completed square form '2x^2+4x+6' "

The only other thing I can think of is to rewrite $\displaystyle 2x^2 + 4x + 6$ as:
$\displaystyle 2x^2 + 4x + 6$

$\displaystyle 2(x^2 + 2x + 3)$

$\displaystyle 2((x^2 + 2x) + 3)$

$\displaystyle 2((x^2 + 2x + 1 - 1) + 3)$

$\displaystyle 2((x^2 + 2x + 1) - 1 + 3)$

$\displaystyle 2((x - 1)^2 + 2)$

$\displaystyle 2(x + 1)^2 + 4$

This method is called "completing the square." Is this what you mean?

-Dan

5. Yes, thats right

So how would something like -x^2+3x+10 be done because Im not sure how to do an equation with a -x^2 instead of a more common x^2.

6. Originally Posted by Yppolitia
Yes, thats right

So how would something like -x^2+3x+10 be done because Im not sure how to do an equation with a -x^2 instead of a more common x^2.
What I always do is factor the coefficient of the $\displaystyle x^2$ term:

$\displaystyle -x^2 + 3x + 10$

$\displaystyle -(x^2 - 3x - 10)$

Then isolate the quadratic and linear x terms:

$\displaystyle -((x^2 - 3x) - 10)$

Then add and subtract the constant that completes the square in the parenthesis. Since the coefficient of the $\displaystyle x^2$ term is always 1, take the linear coefficient (-3 in this case), divide by 2 then square it:

$\displaystyle -\left ( \left ( x^2 - 3x + \frac{9}{4} - \frac{9}{4} \right ) - 10 \right )$

$\displaystyle -\left ( \left ( x^2 - 3x + \frac{9}{4} \right ) - \frac{9}{4} - \frac{40}{4} \right )$

Now simplify:
$\displaystyle -\left ( \left ( x - \frac{3}{2} \right )^2 - \frac{49}{4} \right )$

$\displaystyle - \left ( x - \frac{3}{2} \right )^2 + \frac{49}{4}$

-Dan