If (x − 2)(x − 3) = (a − 2)(a − 3), solve for x A. x = 0 or 5 B. x = 2 or 3 C. x = a or 2 D. x = a or 3 E. x = a or 5 –a

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jeew-m's profile pic

jeew-m | College Teacher | (Level 1) Educator Emeritus

Posted on

`(x-2)(x-3) = (a-2)(a-3)`

`x^2-5x+6 = a^2-5a+6`

`x^2-a^2-5(x-a) = 0`

`(x-a)(x+a)-5(x-a) = 0`

`(x-a)(x+a-5) = 0`

`(x-a)(x-(5-a)) = 0`

`x = a` or `x = 5-a`

So the answers are x = a or x = 5-a.

The correct answer is in E.

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steveschoen | College Teacher | (Level 1) Associate Educator

Posted on

First, multiply each side out.  You would get:

x^2 - 5x + 6 = a^2 - 5a + 6

The 6's cancel out

So, we have:

x^2 - 5x = a^2 - 5a

Get everything on one side.  Subtract a^2 and add 5a, we have:

x^2 - a^2 - 5x + 5a = 0

Or:

(x^2 - a^2) - (5x - 5a) = 0

Factoring each parenthesis:

(x+a)(x-a) - 5(x-a) = 0

Or:

[(x+a)(x-a) - 5(x-a)] = 0

Taking out (x-a), outside the brackets (factor out), we have:

(x-a)[x+a-5] = 0

Setting each parenthesis and bracket = 0 and solving for x, we get:

x-a = 0,     x = a

x+a-5 = 0,     x = -a+5   or   x = 5-a

So, x = a, 5-a.  The answer would be E.

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