If the expression \frac{10x^2 + x - 3}{2x - 1} is equivalent to ax + b for all x \neq 0.5,… — SAT

If the expression $\frac{10x^2 + x - 3}{2x - 1}$ is equivalent to $ax + b$ for all $x \neq 0.5$ , where $a$ and $b$ are constants, what is the value of $a + b$ ?

Answer: 8

Answer

The value of

a + b

is 8.

To find the equivalent expression, factor the numerator:

10x^2 + x - 3 = (2x - 1)(5x + 3)

. For all

x \neq 0.5

, the denominator

2x - 1

is non-zero, so we can divide out the common factor

(2x - 1)

to get

5x + 3

. Matching

5x + 3

with

ax + b

gives

a = 5

and

b = 3

. The sum of these constants is

5 + 3 = 8

Step-by-Step Solution

Factor the numerator of the expression

10x^2 + x - 3 = (2x - 1)(5x + 3)

Factoring the quadratic trinomial allows us to identify common factors that can be simplified.

Simplify the rational expression

\frac{(2x - 1)(5x + 3)}{2x - 1} = 5x + 3

(for

x \neq 0.5

)

Since

x \neq 0.5

, the term

2x - 1

is non-zero and can be canceled from both the numerator and the denominator.

Identify the values of

a

and

b

a = 5

and

b = 3

By comparing the simplified expression

5x + 3

to the form

ax + b

, the coefficients of corresponding terms must be equal.

Calculate the sum of

a

and

b

5 + 3 = 8

The question asks for the sum of the constants

a

and

b

Key Concept

Simplifying rational expressions by factoring and coefficient matching

Question

Answer

Step-by-Step Solution

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