To solve the equation \( x^2 + 28 = -8x \) using the method of completing the square, we can first rearrange it into a more standard form. Here are the steps:
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Start with the original equation: \[ x^2 + 28 = -8x \] We can rearrange it: \[ x^2 + 8x + 28 = 0 \]
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Next, we need to isolate the quadratic and linear terms on one side. To do this, we'll move the constant term to the other side: \[ x^2 + 8x = -28 \]
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Now, we will complete the square. To do this, take half of the coefficient of \( x \) (which is \( 8 \)), square it, and add it to both sides. Half of \( 8 \) is \( 4 \), and squaring it gives \( 16 \): \[ x^2 + 8x + 16 = -28 + 16 \] Simplifying the right side: \[ x^2 + 8x + 16 = -12 \]
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The left side can be factored as a perfect square: \[ (x + 4)^2 = -12 \]
Now we have completed the square, and we obtain the intermediate step: \[ (x + 4)^2 = -12 \]
This is the intermediate step in the form \( (x + a)^2 = b \) where \( a = 4 \) and \( b = -12 \).