Show that
$$\frac{2x^2 + 13x + 20}{2x^2 + x - 10}$$
simplifies to
$$\frac{x + a}{x - b}$$
where a and b are integers. - OCR - GCSE Maths - Question 19 - 2018 - Paper 1
Question 19
Show that
$$\frac{2x^2 + 13x + 20}{2x^2 + x - 10}$$
simplifies to
$$\frac{x + a}{x - b}$$
where a and b are integers.
Worked Solution & Example Answer:Show that
$$\frac{2x^2 + 13x + 20}{2x^2 + x - 10}$$
simplifies to
$$\frac{x + a}{x - b}$$
where a and b are integers. - OCR - GCSE Maths - Question 19 - 2018 - Paper 1
Step 1
Factor the numerator 2x² + 13x + 20
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Answer
To factor the quadratic expression in the numerator, we need to find two numbers that multiply to (2 \cdot 20 = 40) and add up to (13). The numbers (8) and (5) work since (8 + 5 = 13). Therefore, we can factor it as:
2x2+13x+20=(2x+5)(x+4)
Step 2
Factor the denominator 2x² + x - 10
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Answer
For the denominator, we find two numbers that multiply to (2 \cdot (-10) = -20) and add up to (1). The numbers (5) and (-4) satisfy this condition. Thus, the factorization is:
2x2+x−10=(2x−4)(x+5)
Step 3
Simplify the expression
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Answer
Now substituting the factored forms back into the fraction, we have:
(2x−4)(x+5)(2x+5)(x+4)
We can simplify this expression by canceling out common factors:
(2x+5)(x−2)(2x+5)(x+4)=x−2x+4
Thus, we can identify (a = 4) and (b = 2).
