The rational expression \frac{x^2 - 4}{2x^2 + 5x - 3} is undefined for two real values of … — ACT

The rational expression $\frac{x^2 - 4}{2x^2 + 5x - 3}$ is undefined for two real values of $x$ . If the smaller value is $a$ and the larger value is $b$ , what are the values of $a$ and $b$ ?

Answer:

a =

【-3】

b =

【1/2】

Answer

The smaller value is

-3

and the larger value is

1/2

(or

0.5

A rational expression is undefined when the denominator is equal to zero. To find these values, set the denominator

2x^2 + 5x - 3

equal to

0

and solve for

x

. Factoring the quadratic yields

(2x - 1)(x + 3) = 0

. Setting each factor to zero gives

x = 1/2

and

x = -3

. Since

-3

is less than

1/2

, the smaller value is

-3

and the larger value is

1/2

(or

0.5

Step-by-Step Solution

Identify the condition that makes a rational expression undefined.

The rational expression is undefined when its denominator is equal to zero:

2x^2 + 5x - 3 = 0

Division by zero is undefined in the set of real numbers.

Factor the quadratic expression in the denominator.

2x^2 + 5x - 3 = (2x - 1)(x + 3) = 0

Factoring allows us to find the roots of the quadratic equation using the zero product property.

Solve for the roots of the factored equation.

2x - 1 = 0 \Rightarrow x = 1/2

, and

x + 3 = 0 \Rightarrow x = -3

Setting each linear factor to zero determines the values of

x

that make the denominator zero.

Assign the values to the variables based on the inequality constraint.

Since

-3 < 1/2

, the smaller value is

a = -3

and the larger value is

b = 1/2

(or

0.5

The question specifies that

a

is the smaller value and

b

is the larger value.

Key Concept

Identifying domain restrictions of rational expressions by finding where the denominator is equal to zero.

Estimated Time:1m 30s

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