My instructor told us, to solve a quadratic inequality factor out it's roots to set up a test interval on a number line. Then, test to see at what positions the inequality is true.

My question:

$\displaystyle x^2+4x+4\geq 9$
Descartes rule of signs:
0 positive roots
1 negative root

rational root theorem:
$\displaystyle \textrm{roots}=\left [ -4,-2,-1 \right ]$
Synthetic division:
............-4..0
-4......1..4..4..
........1..0..4---- $\displaystyle r=4\,\,\, \therefore$not a root
............-2..-4
-2.......1..4..4
.........1..2..0--------$\displaystyle r=4\,\,\,\therefore$ a root!

root=-2
$\displaystyle (x+2)(x+2)\geq 9$

But, this is a double root.
And because it is a double root my interval is... not applicable besides$\displaystyle \left ( -\infty, -2 \right )$or$\displaystyle \left ( -2,\infty ) \right )$

But my textbook says the answer is:
$\displaystyle \left ( -\infty, -5 \right ]\cup \left [ 1, \infty \right )$

Any help on this problem?

Also, if anyone has a more 'algebraic' way of solving a quadratic inequality, please post it. I hate guess and check problems.

2. Originally Posted by integral
My instructor told us, to solve a quadratic inequality factor out it's roots to set up a test interval on a number line. Then, test to see at what positions the inequality is true.

My question:

$\displaystyle x^2+4x+4\geq 9$
Descartes rule of signs:
0 positive roots
1 negative root

rational root theorem:
$\displaystyle \textrm{roots}=\left [ -4,-2,-1 \right ]$
Synthetic division:
............-4..0
-4......1..4..4..
........1..0..4---- $\displaystyle r=4\,\,\, \therefore$not a root
............-2..-4
-2.......1..4..4
.........1..2..0--------$\displaystyle r=4\,\,\,\therefore$ a root!

root=-2
$\displaystyle (x+2)(x+2)\geq 9$

But, this is a double root.
And because it is a double root my interval is... not applicable besides

But my textbook says the answer is:
$\displaystyle \left ( -\infty, -5 \right ]\cup \left [ 1, \infty \right )$

Any help on this problem?

Also, if anyone has a more 'algebraic' way of solving a quadratic inequality, please post it. I hate guess and check problems.
Hi integral,

You first must solve the quadratic equation:

$\displaystyle x^2+4x+4=9$

$\displaystyle x^2+4x-5=0$

$\displaystyle (x+5)(x-1)=0$

$\displaystyle x=-5 \:\r\:\:x=1$

Mark these two points on a number line. They will be part of your solution.

Check a value in the interval to the left of -5 to see if it makes the original inequality true.

You'll find that it does.

Check a value between -5 and 1 to see if it makes the original inequality true..

You'll find that it doesn't.

Finally, check a value in the interval to the right of 1 to see if it makes the original inequality true.

You'll find that it does.

Conclusion:

Everything from -5 to negative infinity works, including -5.

Everything from 1 to positive infinity works, including 1.

Mathematically:

$\displaystyle \left ( -\infty, -5 \right ]\cup \left [ 1, \infty \right )$