What is the sum of all real values of x that satisfy the equation \frac{x^2 - 3x}{x^2 - 3x… — ACT

What is the sum of all real values of

x

that satisfy the equation

\frac{x^2 - 3x}{x^2 - 3x + 2} + \frac{x^2 - 3x - 1}{x^2 - 3x - 3} = \frac{11}{3}

Answer: 3

Answer

The sum of all real values of

x

that satisfy the equation is

3

Substituting

y = x^2 - 3x

simplifies the original rational equation into the quadratic form

y^2 - y - 12 = 0

. Solving for

y

yields the values

4

and

-3

. Substituting back

x^2 - 3x

for

y

produces two quadratic equations. The first,

x^2 - 3x - 4 = 0

, has real solutions of

4

and

-1

. The second,

x^2 - 3x + 3 = 0

, has a negative discriminant and produces no real solutions. Summing the valid real solutions gives

4 + (-1) = 3

Step-by-Step Solution

Substitute

y = x^2 - 3x

into the equation to simplify the rational terms.

\frac{y}{y+2} + \frac{y-1}{y-3} = \frac{11}{3}

Using a temporary variable simplifies the algebraic manipulation of the rational expressions.

Multiply the entire equation by the least common denominator,

3(y+2)(y-3)

, to eliminate all fractions.

3y(y-3) + 3(y-1)(y+2) = 11(y+2)(y-3)

, where

y \neq -2

and

y \neq 3

This clears the denominators so the equation can be solved as a polynomial.

Expand the terms on both sides of the equation.

(3y^2 - 9y) + (3y^2 + 3y - 6) = 11y^2 - 11y - 66

Expanding allows for combining like terms.

Combine like terms and move all terms to one side of the equation to set it equal to zero.

5y^2 - 5y - 60 = 0

This sets up the expression in the standard quadratic form

ay^2 + by + c = 0

Divide the entire quadratic equation by its greatest common factor,

5

, and factor the resulting expression.

y^2 - y - 12 = 0 \implies (y-4)(y+3) = 0 \implies y = 4 \text{ or } y = -3

Factoring solves for the possible values of the substituted variable

y

Substitute back

y = x^2 - 3x

for each case and solve the resulting quadratic equations for

x

For

y = 4

x^2 - 3x - 4 = 0 \implies (x-4)(x+1) = 0 \implies x = 4 \text{ or } x = -1

.
For

y = -3

x^2 - 3x + 3 = 0

. The discriminant is

(-3)^2 - 4(1)(3) = -3 < 0

, which means there are no real solutions.

This determines the real values of

x

that solve the original equation.

Verify that neither solution makes the original denominators zero, and add the valid real solutions.

4 + (-1) = 3

Since both

x = 4

and

x = -1

result in non-zero denominators, both are valid real solutions. Their sum is

3

Key Concept

Solving rational equations by utilizing algebraic substitution to reduce complexity and analyzing quadratic equations for real solutions.

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