How do you solve the equation #x^2+x+1/4=9/16# by completing the square?

Answer 1

#x=1/4" or "-5/4#

reorganize in the manner shown below

#x^2+x=9/16-1/4#
#x^2+x=5/16#
complete the square #LHS#
#(x^2+x+(1/2)^2)-(1/2)^2=5/16#
#(x+1/2)^2-1/4=5/16#
now solve for #x#
#(x+1/2)^2=5/16+1/4=9/16#
#(x+1/2)^2=9/16#
#x+1/2=+-sqrt(9/16)#
#x+1/2=+-3/4#
#x=-1/2+-3/4#
#x_1=-1/2+3/4=1/4#
#x_2=-1/2-3/4=-5/4#
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Answer 2

To solve the equation (x^2 + x + \frac{1}{4} = \frac{9}{16}) by completing the square, follow these steps:

  1. Move the constant term to the other side of the equation: [x^2 + x = \frac{9}{16} - \frac{1}{4}]

  2. Find the common denominator and combine the fractions on the right side: [x^2 + x = \frac{9}{16} - \frac{4}{16} = \frac{5}{16}]

  3. 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]

  4. Simplify both sides of the equation: [x^2 + x + \frac{1}{4} = \frac{5}{16} + \frac{1}{4}]

  5. Factor the left side of the equation: [\left(x + \frac{1}{2}\right)^2 = \frac{5}{16} + \frac{4}{16}]

  6. Combine the fractions on the right side: [\left(x + \frac{1}{2}\right)^2 = \frac{9}{16}]

  7. Take the square root of both sides: [x + \frac{1}{2} = \pm \sqrt{\frac{9}{16}}]

  8. Simplify the right side: [x + \frac{1}{2} = \pm \frac{3}{4}]

  9. Solve for (x): [x = -\frac{1}{2} \pm \frac{3}{4}]

  10. 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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Answer from HIX Tutor

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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