Pregunta
Solve for \( x \) \[ (x+7)^{2}=2 x^{2}+16 x+25 \] If there is more than one solution, separate them with commas. \( x=\square \)
Ask by Boone Vaughn. in the United States
Feb 25,2025
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\( x = 4, -6 \)
Solución
To solve the equation
(x + 7)² = 2x² + 16x + 25,
follow these steps:
1. Expand the left side:
(x + 7)² = x² + 14x + 49.
2. Set up the equation:
x² + 14x + 49 = 2x² + 16x + 25.
3. Rearrange the equation by subtracting (x² + 14x + 49) from both sides:
0 = 2x² + 16x + 25 - (x² + 14x + 49),
0 = x² + 2x - 24.
4. Solve the quadratic equation x² + 2x - 24 = 0.
Use the quadratic formula:
x = [ -b ± √(b² - 4ac) ] / (2a)
Here, a = 1, b = 2, c = -24.
Calculate the discriminant:
b² - 4ac = (2)² - 4(1)(-24) = 4 + 96 = 100.
Thus,
x = [ -2 ± √100 ] / 2
x = [ -2 ± 10 ] / 2.
5. Find the two solutions:
x = (8)/2 = 4 or x = (-12)/2 = -6.
So, the solutions are x = 4, -6.
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To solve the equation \( (x+7)^{2}=2 x^{2}+16 x+25 \), we first expand the left side: \[ (x + 7)^2 = x^2 + 14x + 49 \] So the equation becomes: \[ x^2 + 14x + 49 = 2x^2 + 16x + 25 \] Next, we rearrange the equation by bringing all terms to one side: \[ x^2 + 14x + 49 - 2x^2 - 16x - 25 = 0 \] This simplifies to: \[ -x^2 - 2x + 24 = 0 \] Multiplying throughout by -1 gives: \[ x^2 + 2x - 24 = 0 \] Now, we can factor this quadratic equation: \[ (x + 6)(x - 4) = 0 \] Setting each factor to zero gives us the solutions: \[ x + 6 = 0 \quad \Rightarrow \quad x = -6 \] \[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \] Thus, the solutions are: \( x = -6, 4 \)
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