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Simplify the expression rac(x+1)(4a2-ax)x2-4ax-4a+x and find two consecutive even numbers whose sum is 270.

Mathematics

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2 parts
12

-a

13

134 and 136134 \text{ and } 136

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Part 12· 3 steps

Simplify the expression (x+1)(4a2−ax)x2−4ax−4a+x\frac{(x+1)(4a^2-ax)}{x^2-4ax-4a+x}.

  1. Factor the numerator.

    Factor out aa from the second term in the numerator:

    (x+1)(4a2−ax)=(x+1)a(4a−x)(x+1)(4a^2-ax) = (x+1)a(4a-x)
  2. Factor the denominator.

    Group terms in the denominator and factor by grouping:

    x2−4ax−4a+x=x2+x−4ax−4ax^2-4ax-4a+x = x^2+x - 4ax-4a =x(x+1)−4a(x+1)= x(x+1) - 4a(x+1) =(x+1)(x−4a)= (x+1)(x-4a)
  3. Simplify the expression.

    Substitute the factored forms back into the expression:

    (x+1)a(4a−x)(x+1)(x−4a)\frac{(x+1)a(4a-x)}{(x+1)(x-4a)}

    Notice that (4a−x)=−(x−4a)(4a-x) = -(x-4a).

    (x+1)a(−(x−4a))(x+1)(x−4a)\frac{(x+1)a(-(x-4a))}{(x+1)(x-4a)}

    Assuming x≠−1x \ne -1 and x≠4ax \ne 4a, we can cancel out the common factors (x+1)(x+1) and (x−4a)(x-4a):

    =−a= -a

    The simplified expression is −a\boxed{-a}.

Part 13

134 and 136134 \text{ and } 136

The sum of two consecutive even numbers is 270. Find the numbers.

  1. Define the two consecutive even numbers.

    Let the first even number be xx.
    The next consecutive even number is x+2x+2.

  2. Set up the equation based on their sum.

    x+(x+2)=270x + (x+2) = 270
  3. Solve for xx.

    2x+2=2702x + 2 = 270 2x=270−22x = 270 - 2 2x=2682x = 268 x=2682x = \frac{268}{2} x=134x = 134
  4. Find the two numbers.

    The first number is x=134x = 134.
    The second number is x+2=134+2=136x+2 = 134+2 = 136.
    The two consecutive even numbers are 134 and 136\boxed{134 \text{ and } 136}.

    **

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