Question

Given that (x,12,2x+3)
is a Pythagorean triple, what is the value of x?(1 point)
Responses

x=9
x equals 9

x=5
x equals 5

x=25
x equals 25

x=13

Answer 1

A Pythagorean triple consists of three positive integers \(a\), \(b\), and \(c\) satisfying the equation \(a^2 + b^2 = c^2\), where \(c\) is the largest number.

In this case, we have the triple \((x, 12, 2x + 3)\). To solve for \(x\), we can assume that \(c = 2x + 3\) (the largest side), and we can write the equation:

\[
x^2 + 12^2 = (2x + 3)^2
\]

Calculating \(12^2\):

\[
12^2 = 144
\]

Now we have:

\[
x^2 + 144 = (2x + 3)^2
\]

Expanding the right side:

\[
(2x + 3)^2 = 4x^2 + 12x + 9
\]

Substituting this back into the equation gives:

\[
x^2 + 144 = 4x^2 + 12x + 9
\]

Now, rearranging the equation to bring all the terms to one side:

\[
x^2 - 4x^2 - 12x + 144 - 9 = 0
\]

This simplifies to:

\[
-3x^2 - 12x + 135 = 0
\]

Multiplying the entire equation by -1 to simplify:

\[
3x^2 + 12x - 135 = 0
\]

Now we can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 3\), \(b = 12\), and \(c = -135\):

Calculating the discriminant:

\[
b^2 - 4ac = 12^2 - 4 \cdot 3 \cdot (-135) = 144 + 1620 = 1764
\]

Now, calculating \(x\):

\[
x = \frac{-12 \pm \sqrt{1764}}{2 \cdot 3}
\]
\[
\sqrt{1764} = 42
\]
\[
x = \frac{-12 \pm 42}{6}
\]

This gives us two possible values for \(x\):

1. \(x = \frac{30}{6} = 5\)
2. \(x = \frac{-54}{6} = -9\) (which we discard since \(x\) must be positive).

Thus, the value of \(x\) is:

\(\boxed{5}\)