How do you solve #x^2=24x+10# by completing the square?

Answer 1

#color(green)(x=12-sqrt(154))color(white)("XX")# or #color(white)("XX")color(green)(x=12+sqrt(154)#

Given #color(white)("XXX")x^2=24x+10#
Shift all the terms which include the variable #x# to the left side: #color(white)("XXX")x^2-24x=10#
Since, in general, the expansion of a squared binomial has the structure: #color(white)("XXX")(x+a)^2=underline(x^2+2ax) + a^2#
If #x^2-24x# are the first two terms of the expansion of a squared binomial then the third terms should be #(-24/2)^2=(-12)^2=12^2#
In order to complete the square we will need to add #12^2# (to both sides)
#color(white)("XXX")x^2-24x+12^2=10+12^2#
Writing as a squared binomial and simplifying: #color(white)("XXX")(x-12)^2=154#
Taking the square root of both sides #color(white)("XXX")x-12=+-sqrt(154)# Adding #12# to both sides #color(white)("XXX")x=12+-sqrt(154)#
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Answer 2

#x=12+-sqrt(154)#

#x^2-24x = 10# take the factor of x, half it and square, then add both sides #x^2-24x+144 = 10+144# #(x-12)^2 = 154# #sqrt((x-12)^2)=+-sqrt(154)# #(x-12)=+-sqrt(154)# #x=12+-sqrt(154)#
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Answer 3

To solve the equation ( x^2 = 24x + 10 ) by completing the square, follow these steps:

  1. Move the constant term to the other side of the equation: [ x^2 - 24x = 10 ]

  2. Take half of the coefficient of the x-term (-24), square it, and add it to both sides of the equation: [ x^2 - 24x + (-24/2)^2 = 10 + (-24/2)^2 ] [ x^2 - 24x + 144 = 10 + 144 ]

  3. Simplify the equation: [ x^2 - 24x + 144 = 154 ]

  4. Write the left side of the equation as a perfect square: [ (x - 12)^2 = 154 ]

  5. Take the square root of both sides and solve for x: [ x - 12 = \pm \sqrt{154} ] [ x = 12 \pm \sqrt{154} ]

Therefore, the solutions to the equation ( x^2 = 24x + 10 ) by completing the square are: [ x = 12 + \sqrt{154} ] and ( x = 12 - \sqrt{154} ).

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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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