Question

How do you solve `1/(x+3)+1/(x+5)=1`?

Answer 1

Resolving this gives: `"x^2+6x+7=0`

I will let you finish that off (you need to use the formula).

Explanation:

The bottom number/expression (denominator) need to be the same to enable direct addition.

Multiply by 1 and you do not change the value. Multiply by 1 but in the form of `1=(x+5)/(x+5)` you do not change the value but you do change the way it looks.

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Multiply `1/(x+3)` by 1 but in the form of `1=(x+5)/(x+5)`

Giving `color(brown)(1/(x+3)xx(x+5)/(x+5) = (x+5)/((x+3)(x+5)))`

Multiply `1/(x+5)` by 1 but in the form of `1=(x+3)/(x+3)`

Giving `color(brown)(1/(x+5)xx(x+3)/(x+3) = (x+3)/((x+3)(x+5)))`

`color(blue)("Putting it all together")`

`" "(x+5)/((x+3)(x+5)) + (x+3)/((x+3)(x+5))=1`

`" "(2x+8)/((x+3)(x+5))=1`

Multiply both sides by `(x+3)(x+5)` giving

`" "(2x+8)xx(cancel((x+3)(x+5)))/(cancel((x+3)(x+5)))=1xx(x+3)(x+5)`

`" "2x+8=x^2+8x+15`

`" "x^2+6x+7=0`

Use the formula to solve for `x`

I will let you do that

`y=ax^2+bx+c" "` where `" "x=(-b+-sqrt(b^2-4ac))/(2a)`
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