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ACT Math : How to find the solution of a rational equation with a binomial denominator

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Example Questions

Example Question #1 : Binomial Denominators

For the equation $x^{2}-10x+21=0$ $\displaystyle x^{2}-10x+21=0$ , what is(are) the solution(s) for $\displaystyle x$ ?

Possible Answers:

2,5 $\displaystyle 2,5$

$\displaystyle -7$

3,7 $\displaystyle 3,7$

$\displaystyle -3$

-3,-7 $\displaystyle -3,-7$

Correct answer:

3,7 $\displaystyle 3,7$

Explanation:

$x^{2}-10x+21=0$ $\displaystyle x^{2}-10x+21=0$ , can be factored to (x -7)(x-3) = 0. Therefore, x-7 = 0 and x-3 = 0. Solving for x in both cases, gives 7 and 3.

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Example Question #52 : Variables

Simplify:

$\frac{x^2+3x+4x+12}{x+4}$ $\displaystyle \frac{x^2+3x+4x+12}{x+4}$

Possible Answers:

x-4 $\displaystyle x-4$

$\frac{x^2+7x+12}{x+4}$ $\displaystyle \frac{x^2+7x+12}{x+4}$

x-3 $\displaystyle x-3$

x+4 $\displaystyle x+4$

x+3 $\displaystyle x+3$

Correct answer:

x+3 $\displaystyle x+3$

Explanation:

In order to begin this kind of a problem, it's key to look at parts of the rational expression that can be simplified.

In this case, the denominator is an already-simplified binomial; however, the numerator can be factored through "factoring by grouping." This can be a helpful idea to keep in mind when you come across a polynomial with four terms and simplifying is involved.

${x^2+3x+4x+12}$ $\displaystyle {x^2+3x+4x+12}$ can be simplified first by removing the common factor of $\displaystyle x$ from the first two terms and the common factor of $\displaystyle 4$ from the last two terms:

$\displaystyle x(x+3)+4(x+3)$

This leaves two terms that are identical (x+3) $\displaystyle (x+3)$ and their coefficients, which can be combined into another term to complete the factoring:

$\displaystyle (x+4)(x+3)$

Consider the denominator; the quantity (x+4) $\displaystyle (x+4)$ appears, so the (x+4) $\displaystyle (x+4)$ in the numerator and in the denominator can be cancelled out. The simplified expression is then left as (x+3) $\displaystyle (x+3)$ .

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Example Question #51 : Variables

Simplify:

$\frac{x^2+9x-10}{x-1}$ $\displaystyle \frac{x^2+9x-10}{x-1}$

Possible Answers:

(x+1) $\displaystyle (x+1)$

(x-10) $\displaystyle (x-10)$

(x+2) $\displaystyle (x+2)$

(x+10) $\displaystyle (x+10)$

(x-1) $\displaystyle (x-1)$

Correct answer:

(x+10) $\displaystyle (x+10)$

Explanation:

In order to begin this kind of a problem, it's key to look at parts of the rational expression that can be simplified.

In this case, the denominator is an already-simplified binomial; however, the numerator can be factored.

$\displaystyle x^2+9x-10$

The roots will be numbers that sum up to $\displaystyle 9$ but have the product of $\displaystyle -10$ .

The options include:

$\displaystyle (-1)(10) = -10$
$\displaystyle (1)(-10) = -10$
$\displaystyle (2)(-5) = -10$
$\displaystyle (-2)(5)= -10$

When these options are summed up:

-1+10=9 $\displaystyle -1+10=9$
1-10=-9 $\displaystyle 1-10=-9$
2-5=-3 $\displaystyle 2-5=-3$
-2+5=3 $\displaystyle -2+5=3$

We can negate the last three options because the first option of $\displaystyle -1$ and $\displaystyle 10$ fulfill the requirements. Therefore, the numerator can be factored into the following:

${x^2+9-10}=(x-1)(x+10)$ $\displaystyle {x^2+9-10}=(x-1)(x+10)$

Because the quantity (x-1) $\displaystyle (x-1)$ appears in the denominator, this can be "canceled out." This leaves the final answer to be the quantity (x+10) $\displaystyle (x+10)$ .

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ACT Math : How to find the solution of a rational equation with a binomial denominator

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : Binomial Denominators

Example Question #52 : Variables

Example Question #51 : Variables

All ACT Math Resources

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