Given that b=(2−9),c=(1x), a=7b+7c and the magnitude of a is 3577−−−−√, find x.

We are given the following information:

- \( b = (2 - 9) \)
- \( c = (1x) \)
- \( a = 7b + 7c \)
- The magnitude of \( a \) is \( \sqrt{3577} \)

### Step 1: Compute \( 7b + 7c \)

First, multiply both \( b \) and \( c \) by 7:
\[
7b = 7(2 - 9) = (14 - 63)
\]
\[
7c = 7(1x) = (7x)
\]

Thus, \( a \) is:
\[
a = (14 - 63) + (7x) = (14 + 7, -63 + 7x)
\]
\[
a = (21, 7x - 63)
\]

### Step 2: Find the magnitude of \( a \)

The magnitude of \( a \) is given by:
\[
|a| = \sqrt{(21)^2 + (7x - 63)^2}
\]
We are also told that the magnitude of \( a \) is \( \sqrt{3577} \). Therefore:
\[
\sqrt{(21)^2 + (7x - 63)^2} = \sqrt{3577}
\]

Square both sides to eliminate the square roots:
\[
(21)^2 + (7x - 63)^2 = 3577
\]
\[
441 + (7x - 63)^2 = 3577
\]
\[
(7x - 63)^2 = 3577 - 441
\]
\[
(7x - 63)^2 = 3136
\]

### Step 3: Solve for \( x \)

Take the square root of both sides:
\[
7x - 63 = \pm \sqrt{3136}
\]
\[
7x - 63 = \pm 56
\]

Now, solve for \( x \):

#### Case 1: \( 7x - 63 = 56 \)
\[
7x = 56 + 63
\]
\[
7x = 119
\]
\[
x = \frac{119}{7} = 17
\]

#### Case 2: \( 7x - 63 = -56 \)
\[
7x = -56 + 63
\]
\[
7x = 7
\]
\[
x = \frac{7}{7} = 1
\]

### Conclusion:

The possible values of \( x \) are \( x = 17 \) and \( x = 1 \).

Would you like further details or clarifications?

Here are 5 related questions you might find interesting:
1. How can you interpret the direction of vector \( a \) geometrically?
2. What happens if the magnitude of \( a \) is changed to a different value?
3. How would the solution change if \( b \) and \( c \) were in three dimensions?
4. Can you derive a general formula for vectors of this type?
5. What if the components of \( b \) and \( c \) were parametric functions of time?

**Tip:** Always double-check your calculations when working with square roots and solving quadratic equations.

Given that b=(2−9), c=(1x), a=7b+7c and the magnitude of a is sqrt(3577), find x.

Learn how to solve the problem of finding x when given vector b=(2−9), c=(1x), and the magnitude of a=7b+7c is sqrt(3577). This solution involves vector algebra, the magnitude formula, and solving a quadratic equation. Suitable for students in Grades 10-12.

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