Math  /  Algebra

Questionx510x11=0x^{5} - 10x - 11 = 0
25501251,36\frac{25-501}{25-1,-36}
1) 5,362) 3) 4) \begin{array}{l} \text{1) } 5,-36 \\ \text{2) } \\ \text{3) } \\ \text{4) } \end{array}
5)
b)
7)
8)
(9)
8 (1)
(1)

Studdy Solution

STEP 1

What is this asking? Find a value of xx that makes the equation x510x11=0x^5 - 10x - 11 = 0 true. Watch out! This isn't a quadratic equation, so our usual tricks won't work!
We'll need to be clever!

STEP 2

1. Strategize!
2. Test a few values.
3. Verify the solution.

STEP 3

Alright, let's **think** about this equation: x510x11=0x^5 - 10x - 11 = 0.
It looks intimidating, but we can totally handle it!
We're looking for a value of xx that makes the left side equal to **zero**.

STEP 4

Since we don't have a direct way to solve this, let's try some **strategic guessing**!
We want a number that, when multiplied by itself five times (x5x^5), is close to 1010 times itself plus 1111.

STEP 5

Let's start by testing x=1x = 1.
Plugging that in, we get: 1510111=11011=201^5 - 10 \cdot 1 - 11 = 1 - 10 - 11 = -20 Hmm, 20-20 is pretty far from zero.

STEP 6

Let's try x=2x = 2.
Substituting x=2x = 2 into the equation gives us: 2510211=322011=12^5 - 10 \cdot 2 - 11 = 32 - 20 - 11 = 1 Wow, 11 is much closer to zero!
We're getting warmer!

STEP 7

Since x=2x = 2 gave us a result slightly greater than zero, and x=1x = 1 gave us a result much less than zero, let's try a value between 11 and 22.
But, since we're dealing with x5x^5, let's try something closer to 22.

STEP 8

Let's try x=2.1x = 2.1.
Substituting x=2.1x = 2.1 into the equation gives us: (2.1)510(2.1)11=40.841012111=8.84101(2.1)^5 - 10 \cdot (2.1) - 11 = 40.84101 - 21 - 11 = 8.84101 Oops, we overshot a bit!

STEP 9

Let's go back a little and try x=2.05x = 2.05.
Substituting x=2.05x = 2.05 into the equation gives us: (2.05)510(2.05)11=36.936420.511=5.4364(2.05)^5 - 10 \cdot (2.05) - 11 = 36.9364 - 20.5 - 11 = 5.4364 Still a bit high.

STEP 10

Let's try x=2.01x = 2.01.
Substituting x=2.01x = 2.01 into the equation gives us: (2.01)510(2.01)11=32.622220.111=1.4222(2.01)^5 - 10 \cdot (2.01) - 11 = 32.6222 - 20.1 - 11 = 1.4222 Getting closer!

STEP 11

Let's try x=2.001x = 2.001.
Substituting x=2.001x = 2.001 into the equation gives us: (2.001)510(2.001)11=32.064020.0111=1.054(2.001)^5 - 10 \cdot (2.001) - 11 = 32.0640 - 20.01 - 11 = 1.054 Even closer!

STEP 12

Notice that the other stuff on the page seems like a distraction!
Let's try a whole number again.
What about x=1x = -1? (1)510(1)11=1+1011=2(-1)^5 - 10 \cdot (-1) - 11 = -1 + 10 - 11 = -2 Getting closer to zero again!

STEP 13

Let's try x=1.1x = -1.1. (1.1)510(1.1)11=1.61051+1111=1.61051(-1.1)^5 - 10 \cdot (-1.1) - 11 = -1.61051 + 11 - 11 = -1.61051

STEP 14

Let's try x=1.2x = -1.2. (1.2)510(1.2)11=2.48832+1211=1.48832(-1.2)^5 - 10 \cdot (-1.2) - 11 = -2.48832 + 12 - 11 = -1.48832

STEP 15

Let's try x=1.3x = -1.3. (1.3)510(1.3)11=3.71293+1311=1.71293(-1.3)^5 - 10 \cdot (-1.3) - 11 = -3.71293 + 13 - 11 = -1.71293

STEP 16

Let's try x=1.01x = -1.01. (1.01)510(1.01)11=1.05101+10.111=2.0(-1.01)^5 - 10 \cdot (-1.01) - 11 = -1.05101 + 10.1 - 11 = -2.0 It looks like we are getting further away from zero.

STEP 17

Let's try x=0.9x = -0.9. (0.9)510(0.9)11=0.59049+911=2.59049(-0.9)^5 - 10 \cdot (-0.9) - 11 = -0.59049 + 9 - 11 = -2.59049

STEP 18

Let's try x=0x = 0. (0)510(0)11=11(0)^5 - 10 \cdot (0) - 11 = -11

STEP 19

Let's try x=2x = -2. (2)510(2)11=32+2011=23(-2)^5 - 10 \cdot (-2) - 11 = -32 + 20 - 11 = -23

STEP 20

It seems like x=2x = 2 gives us the closest result to zero so far.

STEP 21

Although x=2x = 2 isn't a perfect solution, it's the closest we've gotten.
Let's call it our **approximate solution**.

STEP 22

Our approximate solution is x2x \approx 2.

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