Find the value of x if the given distance between points (-2, 5) and (x, 19) be Square root 205.

To find the value of \( x \), we will use the distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \):

\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Given points are \( (-2, 5) \) and \( (x, 19) \), and the distance between them is \( \sqrt{205} \).

Substitute into the distance formula:

\[
\sqrt{205} = \sqrt{(x - (-2))^2 + (19 - 5)^2}
\]

Simplify the equation:

\[
\sqrt{205} = \sqrt{(x + 2)^2 + (19 - 5)^2}
\]
\[
\sqrt{205} = \sqrt{(x + 2)^2 + 14^2}
\]
\[
\sqrt{205} = \sqrt{(x + 2)^2 + 196}
\]

Now, square both sides to remove the square roots:

\[
205 = (x + 2)^2 + 196
\]

Subtract 196 from both sides:

\[
205 - 196 = (x + 2)^2
\]
\[
9 = (x + 2)^2
\]

Take the square root of both sides:

\[
x + 2 = \pm 3
\]

Now, solve for \( x \):

1. \( x + 2 = 3 \Rightarrow x = 1 \)
2. \( x + 2 = -3 \Rightarrow x = -5 \)

Thus, the possible values of \( x \) are \( x = 1 \) and \( x = -5 \).

Would you like further details, or do you have any questions?

Here are 5 related questions to explore:

1. What is the general distance formula between two points?
2. How do we find the midpoint between two points?
3. What is the equation of a circle given a center and radius?
4. How can we derive the slope between two points?
5. How would the formula change if the distance between points were in 3D space?

**Tip**: Always check both the positive and negative roots when solving quadratic equations!

Solve for x in the equation derived from the distance formula between points (-2, 5) and (x, 19), where the distance is given as √205. Learn step-by-step how to apply the distance formula and quadratic equation techniques. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation