SOLUTION: sqrt(2x-1) - sqrt(x+3) = 1

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Question 64609: sqrt(2x-1) - sqrt(x+3) = 1
Answer by praseenakos@yahoo.com(507) (Show Source):
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QUESTION:

sqrt(2x-1) - sqrt(x+3) = 1

ANSWER:
sqrt(2x-1) - sqrt(x+3) = 1

Squaring on both sides,

(sqrt(2x-1) - sqrt(x+3))^2 = 1^2

(2x-1) + (x+3)- 2 sqrt[ (2x-1)(x+3) ] = 1

2x-1 + x+3 - 2 sqrt[ 2x*x + 2x* 3- 1 * x -1 * 3 ] = 1

2x + x - 1 + 3 - 2 sqrt[2x^2 + 6x - x - 3 ] = 1

3x + 2 - 2sqrt [2x^2 + 5x -3] = 1

3x + 2 - 1 = 2sqrt [2x^2 + 5x -3]

3x + 1 = 2sqrt [2x^2 + 5x -3]

Squaring again,

(3x + 1 )^2 = (2sqrt [2x^2 + 5x -3] )^2

9x^2 + 6x + 1 = 4 [2x^2 + 5x -3]

9x^2 + 6x + 1 = 4 * 2x^2 + 4 *5x - 4*3

9x^2 + 6x + 1 = 8x^2 + 20x - 12

9x^2 -8x^2 + 6x - 20x + 1 + 12 = 0

1x^2 - 14x + 13 = 0

x^2 - 14x + 13 = 0

This is a quadratic equation,

For solving this equation we have different methods - 1)Using quadratic formula 2) Splitting middle term.

Splitting middile term:
x^2 - 14x + 13 = 0

Here we have to find out two numbers whose sum is -14 and product is +13.
Such two numbers are -13 and -1

==> x^2 - 13x - 1x+ 13 = 0

==> ( x^2 - 13x )- (1x - 13 ) = 0

==> x (x -13) -1(x-13) = 0

Take out common terms.

==> (x-13)(x-1) = 0

==> either (x-13)= 0 or (x-1) = 0

==> x = 13 or x = 1

So the solution is x = 13 or x = 1

2. Quadratic formula:

The general form of a quadratic equation is ax^2 + bx + c = 0 and its solution is given by the formula,

______________________(1)

Here we have a = 1, b = -14 and c = 13

Substitute these values in (1)

We have,

==>

==>

==>

==> x = (14 + 12 )/ 2 or x = (14 - 12 )/ 2

==> x = 26/2 or x = 2/2

==> x = 13 or x = 1

In both cases we can see that answers are same.

Hope you understood.

Regards.

Praseena.