How do you solve the equation #x^2+x+1/4=9/16# by completing the square?
reorganize in the manner shown below
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To solve the equation (x^2 + x + \frac{1}{4} = \frac{9}{16}) by completing the square, follow these steps:
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Move the constant term to the other side of the equation: [x^2 + x = \frac{9}{16} - \frac{1}{4}]
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Find the common denominator and combine the fractions on the right side: [x^2 + x = \frac{9}{16} - \frac{4}{16} = \frac{5}{16}]
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Complete the square by adding and subtracting (\left(\frac{1}{2} \cdot 1\right)^2) to the left side of the equation: [x^2 + x + \left(\frac{1}{2}\right)^2 = \frac{5}{16} + \left(\frac{1}{2}\right)^2]
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Simplify both sides of the equation: [x^2 + x + \frac{1}{4} = \frac{5}{16} + \frac{1}{4}]
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Factor the left side of the equation: [\left(x + \frac{1}{2}\right)^2 = \frac{5}{16} + \frac{4}{16}]
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Combine the fractions on the right side: [\left(x + \frac{1}{2}\right)^2 = \frac{9}{16}]
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Take the square root of both sides: [x + \frac{1}{2} = \pm \sqrt{\frac{9}{16}}]
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Simplify the right side: [x + \frac{1}{2} = \pm \frac{3}{4}]
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Solve for (x): [x = -\frac{1}{2} \pm \frac{3}{4}]
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Simplify the expressions: [x = -\frac{1}{2} + \frac{3}{4} \quad \text{or} \quad x = -\frac{1}{2} - \frac{3}{4}]
[x = \frac{1}{4} \quad \text{or} \quad x = -\frac{5}{4}]
So, the solutions to the equation (x^2 + x + \frac{1}{4} = \frac{9}{16}) are (x = \frac{1}{4}) and (x = -\frac{5}{4}).
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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