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20 (a) Prove that (2x + 1)(3x + 2) + -(3x + 5) + 2 is a perfect square - OCR - GCSE Maths - Question 20 - 2018 - Paper 5
Question 20
20 (a) Prove that (2x + 1)(3x + 2) + -(3x + 5) + 2 is a perfect square. (b) Gemma says The equation (2x + 1)(3x + 2) - x(3x + 5) + 2 = -12 has no solutions. Explai... show full transcript
Worked Solution & Example Answer:20 (a) Prove that (2x + 1)(3x + 2) + -(3x + 5) + 2 is a perfect square - OCR - GCSE Maths - Question 20 - 2018 - Paper 5
Step 1
(a) Prove that (2x + 1)(3x + 2) + -(3x + 5) + 2 is a perfect square.
Answer
To prove that
is a perfect square, we first need to expand the expression.
-
Expand the Terms:
Now substitute this result back into the initial equation, giving us:
Simplifying further, we have:
-
Rearranging the Expression:
We rewrite it as:
which is a perfect square.
Thus, we have shown that the expression is indeed a perfect square.
Step 2
(b) Explain Gemma's reasoning.
Answer
Gemma's reasoning is based on the properties of quadratic equations. She states that the equation:
has no solutions. This is due to the fact that when rearranging the equation, if we set it equal to zero, we will end up with a negative discriminant, which indicates that there are no real solutions for the variable x. Specifically, if we manipulate the equation and isolate it to standard form, we might find a situation like:
Thus, if the discriminant is less than zero, as Gemma suggests, then the quadratic equation has no real solutions.