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: 2x^2 + 13x + 20
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Answer
To factor the quadratic expression, we look for two numbers that multiply to (2 \times 20 = 40) and add up to 13. The numbers 8 and 5 work:
2x2+8x+5x+20=2x(x+4)+5(x+4)=(2x+5)(x+4)
Step 2
Factor the denominator: 2x^2 + x - 10
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Answer
Next, we need to factor the denominator. We look for two numbers that multiply to (2 \times (-10) = -20) and add up to 1. The numbers 5 and -4 work:
2x2+5x−4x−10=(2x−5)(x+2)
Step 3
Simplify the expression
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Answer
Now substituting the factored forms into our expression, we get:
(2x−5)(x+2)(2x+5)(x+4)
To write it in the required form, we notice that we need to express it as (\frac{x + a}{x - b}). If we let (a = 4) and (b = -2), the expression can be simplified as required.
