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Question

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

(1 point)
Responses

x=25x is equal to 25

x=9x is equal to 9

x=13x is equal to 13

x=5

GPT-4o mini · Answer

To determine the value of $ x $ such that $ (x, 12, 2x + 3) $ forms a Pythagorean triple, we apply the Pythagorean theorem: $$ a^2 + b^2 = c^2 $$ Here, we can identify: - $ a = x $ - $ b = 12 $ - $ c = 2x + 3 $ According to the Pythagorean theorem, we have: $$ x^2 + 12^2 = (2x + 3)^2 $$ Now we can break down the equation step by step. 1. Calculate $ 12^2 $: $$ 12^2 = 144 $$ 2. Expand $ (2x + 3)^2 $: $$ (2x + 3)^2 = 4x^2 + 12x + 9 $$ 3. Now substitute back into the equation: $$ x^2 + 144 = 4x^2 + 12x + 9 $$ 4. Rearranging the equation gives: $$ x^2 - 4x^2 - 12x + 144 - 9 = 0 $$ This simplifies to: $$ -3x^2 - 12x + 135 = 0 $$ 5. Rearranging gives: $$ 3x^2 + 12x - 135 = 0 $$ 6. Dividing the entire equation by 3 simplifies it to: $$ x^2 + 4x - 45 = 0 $$ 7. Now we solve this quadratic equation using the factorization method. We need two numbers that multiply to -45 and add to 4. These numbers are 9 and -5. Thus, we can factor the quadratic as: $$ (x + 9)(x - 5) = 0 $$ Setting each factor equal to zero gives: $$ x + 9 = 0 \quad \text{or} \quad x - 5 = 0 $$ So, $ x = -9 $ or $ x = 5 $. Since x must be a positive value in a Pythagorean triple, we choose: $$ x = 5 $$ Therefore, the value of $ x $ is $ 5 $. The answer is: **x = 5**.