Solve the equation given by completing the square. \( -5 x^{2}-50 x-120=0 \) [Hint: Divide by -5 first] \( x= \)
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Solución
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The Deep Dive
To solve the equation \( -5x^2 - 50x - 120 = 0 \), we first divide everything by -5: \[ x^2 + 10x + 24 = 0 \] Next, we rearrange the equation to isolate the \( x^2 + 10x \) part: \[ x^2 + 10x = -24 \] Now, we complete the square. To do this, take half of the coefficient of \( x \) (which is 10), square it (getting 25), and add it to both sides: \[ x^2 + 10x + 25 = -24 + 25 \] This simplifies to: \[ (x + 5)^2 = 1 \] Taking the square root of both sides gives: \[ x + 5 = \pm 1 \] Now, solving for \( x \): 1. \( x + 5 = 1 \) leads to \( x = -4 \) 2. \( x + 5 = -1 \) leads to \( x = -6 \) Thus, the solutions are: \[ x = -4 \quad \text{and} \quad x = -6 \]
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