Given that $x > 0$ and $x \neq 25$, fully simplify
$$
\frac{10 + 5x - 2x^{2} - \sqrt{x^{3}}}{5 - \sqrt{x}}\n$$
Fully justify your answer. - AQA - A-Level Maths Mechanics - Question 6 - 2021 - Paper 3
Question 6
Given that $x > 0$ and $x \neq 25$, fully simplify
$$
\frac{10 + 5x - 2x^{2} - \sqrt{x^{3}}}{5 - \sqrt{x}}\n$$
Fully justify your answer.
Worked Solution & Example Answer:Given that $x > 0$ and $x \neq 25$, fully simplify
$$
\frac{10 + 5x - 2x^{2} - \sqrt{x^{3}}}{5 - \sqrt{x}}\n$$
Fully justify your answer. - AQA - A-Level Maths Mechanics - Question 6 - 2021 - Paper 3
Step 1
Step 1: Begin to Solve the Problem
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Answer
To simplify the expression, we first write the problem clearly:
5−x10+5x−2x2−x3.
Next, we can observe the numerator. Rewriting it allows us to combine like terms.
Step 2
Step 2: Factor the Numerator
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Answer
The numerator can be approached by rearranging terms:
10+5x−2x2−x3=−2x2+5x+10−x3.
It may help to group terms and factor accordingly. Let's look for common factors.
Step 3
Step 3: Simplify the Expression
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Answer
To simplify further, we can factor out −1 from certain terms to reveal potential common factors:
=−1(2x2−5x−10+x3).
Now, we can inspect the denominator. Notice that we will multiply both the numerator and denominator by the conjugate 5+x to rationalize the denominator.
Step 4
Step 4: Final Simplification
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