Question $y = -6^{4}$
2. $y \ge 2x^2 + 9x + 10$ $4 \cdot 9$ $24 - 9 - 9 + 10$

Studdy Solution

STEP 1

What is this asking? We need to evaluate $-6^4$ and also figure out what values of $x$ make $y \ge 2x^2 + 9x + 10$ true.
Plus, we have a quick multiplication $4 \cdot 9$ and an addition/subtraction problem $24 - 9 - 9 + 10$ . Watch out! Don't forget that $-6^4$ means the negative of $6^4$ , not $(-6)^4$ !
Also, be careful with that inequality; we might need to do some factoring or use the quadratic formula.

STEP 2

1. Evaluate the Power
2. Solve the Inequality
3. Quick Multiplication
4. Simple Arithmetic

STEP 3

Let's tackle $-6^4$ .
Remember, this means we take $6$ to the power of $4$ first and then make it negative.

STEP 4

So, $6^4$ is $6 \cdot 6 \cdot 6 \cdot 6$ .
Let's break it down: $6 \cdot 6 = 36$ , and $36 \cdot 36 = \mathbf{1296}$ .
Therefore, $6^4 = \mathbf{1296}$ .

STEP 5

Now, slap on that negative sign, and we get $-6^4 = \mathbf{-1296}$ .
Boom!

STEP 6

We're looking at $y \ge 2x^2 + 9x + 10$ .
Since we already know $y = -1296$ from the previous step, let's substitute that in: $-1296 \ge 2x^2 + 9x + 10$ .

STEP 7

Let's move everything to one side to get a standard quadratic form.
Adding $1296$ to both sides gives us $0 \ge 2x^2 + 9x + 1306$ .

STEP 8

This quadratic doesn't look easily factorable, so let's use the quadratic formula!
Remember, for $ax^2 + bx + c = 0$ , $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .

STEP 9

In our case, $a = 2$ , $b = 9$ , and $c = 1306$ .
Plugging those values in, we get $x = \frac{-9 \pm \sqrt{9^2 - 4 \cdot 2 \cdot 1306}}{2 \cdot 2}$ .

STEP 10

Simplifying, $x = \frac{-9 \pm \sqrt{81 - 10448}}{4}$ , which becomes $x = \frac{-9 \pm \sqrt{-10367}}{4}$ .

STEP 11

Uh oh!
We have a negative number under the square root.
That means there are no real solutions for $x$ .
So, there are no real values of $x$ that satisfy the inequality.

STEP 12

$4 \cdot 9$ is just $9 + 9 + 9 + 9 = 36$ .
Easy peasy!

STEP 13

Let's break down $24 - 9 - 9 + 10$ .
First, $24 - 9 = 15$ .
Then, $15 - 9 = 6$ .
Finally, $6 + 10 = 16$ .
So, $24 - 9 - 9 + 10 = \mathbf{16}$ .

STEP 14

We found $-6^4 = \mathbf{-1296}$ .
There are no real solutions for $x$ in the inequality $y \ge 2x^2 + 9x + 10$ when $y = -1296$ .
We also calculated $4 \cdot 9 = \mathbf{36}$ and $24 - 9 - 9 + 10 = \mathbf{16}$ .

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