What is the simplified form of x minus 5 over x squared minus 3x minus 10 ⋅ x plus 4 over x squared plus x minus 12? (6 points)

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What is the simplified form of x minus 5 over x squared minus 3x minus 10 ⋅ x plus 4 over x squared plus x minus 12? (6 points)

The correct answer is C: 1/(x + 4)(x - 5)

Why? Well, let's first simplify x^2 - 3x - 10 and x^2 + x - 12; the two denominators. Each of these should become an (x +/- number), and to figure out the number, as well as whether it is positive or negative, we can do a simple trick

Look at the factors of the right number (In this case, -10 and -12)

-10 -12

1 * -10 1 * -12

-1 * 10 -1 * 12

-2 * 5 -2 * 6

2 * -5 2 * -6

-3 * 4

4 * -3

Now for part 2

Which of these pairs add up to the middle number? One of the pairs of -10 should make -3, and likewise, one of the pairs of -12 should make 1 (when x has no number in front of it you may safely assume it is 1).

2 - 5 = -3 and 4 - 3 = 1, so we now know that the 2 fraction equations simplified is

x + 2 / (x + 2)(x - 5) * x - 3/(x -3)(x + 4)

Notice anything repeating? As long as they are apart of the same fraction, we can cross out anything that has the same x - number. Crossing out both x + 2 and x - 3, we now simply have x - 5 * x + 4. Because we crossed out both numerators, the top numbers both become 1, thus giving our answer, C.

[tex]\ \text{a. }\quad\dfrac{1}{(x-3)(x+4)}[/tex]

Step-by-step explanation:

Maybe you want the product ...

[tex]\dfrac{x-5}{x^2-3x-10}\cdot\dfrac{x+2}{x^2+x-12}=\dfrac{x-5}{(x-5)(x+2)}\cdot\dfrac{x+2}{(x-3)(x+4)}\\\\=\boxed{\dfrac{1}{(x-3)(x+4)}}[/tex]

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Numerator factors of (x-5) and (x+2) cancel those in the denominator.

Step 1 is to factor everything!

After factoring you will get:

x-5 over (x+2)(x+5) times x+4 over (x-3)(x+4)

Step 2 is to remove the duplicate factors!

Remove x+4 and x+5

x-5 over (x+2)(x-5) times 1 over x-3 = 1 over x+2 times 1 over x-3

Step 3 is to multiply the numerators and the denominators.

End result is 1 over (x+2)(x-3)

Our simplified form will be

[tex]\frac{1}{(x+2)(x-3)}[/tex]

Step-by-step explanation:

Since we have given that

[tex]\frac{x-5}{x^2-3x-10}\times \frac{x+4}{x^2+x-12}[/tex]

So, first we split the middle term of the denominator ,

[tex]x^2-3x-10=0\\\\=x^2-5x+2x-10\\\\=x(x-5)+2(x-5)\\\\=(x-5)(x+2)[/tex]

Similarly,

[tex]x^2+x-12=0\\\\x^2+4x-3x-12=0\\\\x(x+4)-3(x+4)=0\\\\(x+4)(x-3)=0[/tex]

Now, we put in our expression above:

[tex]\frac{x-5}{(x-5)(x+2)}\times \frac{x+4}{(x-3)(x+4)}\\\\=\frac{1}{x+2}\times \frac{1}{x-3}\\\\=\frac{1}{(x+2)(x-3)}[/tex]

Hence, our simplified form will be

[tex]\frac{1}{(x+2)(x-3)}[/tex]

We can simplify the expression with patience and a little mathematical tinkering.

The simplified expression is shown in the attached picture, along with more information obtained about your expression.

The objective is to add similar terms and fractions and group them together to form a solution in a compact, elegant form.

[tex]What is the simplified form of x minus 5 over x squared minus 3x minus 10 ⋅ x plus 2 over x squared[/tex]

[tex]What is the simplified form of x minus 5 over x squared minus 3x minus 10 ⋅ x plus 2 over x squared[/tex]