What is the sum of all distinct real solutions to the equation \frac{x^2 - 8x + 15}{x - 3}… — GMAT

What is the sum of all distinct real solutions to the equation

\frac{x^2 - 8x + 15}{x - 3} = (x - 5)^2

Factoring the numerator

x^2 - 8x + 15

gives

(x - 3)(x - 5)

. For all

x \neq 3

, the original equation simplifies to

x - 5 = (x - 5)^2

. Rearranging into standard factored form

(x - 5)(x - 6) = 0

yields two distinct real solutions:

x = 5

and

x = 6

. Neither value violates the restriction

x \neq 3

. Adding these values together yields

5 + 6 = 11

Determine the domain restriction for the rational expression

The expression

\frac{x^2 - 8x + 15}{x - 3}

requires that the denominator

x - 3 \neq 0

, so

x \neq 3

Division by zero is undefined in real numbers.

Factor the numerator and simplify the left side of the equation

Since

x^2 - 8x + 15 = (x - 3)(x - 5)

, for

x \neq 3

\frac{(x - 3)(x - 5)}{x - 3} = x - 5

Canceling the common non-zero factor

(x - 3)

simplifies the rational equation.

Solve the simplified quadratic equation for

x

x - 5 = (x - 5)^2 \implies (x - 5)^2 - (x - 5) = 0 \implies (x - 5)(x - 5 - 1) = 0 \implies (x - 5)(x - 6) = 0

. Thus,

x = 5

x = 6

Factoring out the common factor

(x - 5)

preserves all valid solutions.

Verify solutions against the domain restriction and calculate the sum

Both

x = 5

and

x = 6

satisfy

x \neq 3

. The sum of all distinct real solutions is

5 + 6 = 11

Combining the valid distinct roots gives the required total.

Polynomial factoring, domain restrictions on rational expressions, and avoiding root loss when solving quadratic equations.

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