The curve C has equation $y = x^2 + 3x - 3$ - Edexcel - GCSE Maths - Question 16 - 2021 - Paper 3
Question 16
The curve C has equation $y = x^2 + 3x - 3$.
The line L has equation $y - 5x + 4 = 0$.
Show, algebraically, that C and L have exactly one point in common.
Worked Solution & Example Answer:The curve C has equation $y = x^2 + 3x - 3$ - Edexcel - GCSE Maths - Question 16 - 2021 - Paper 3
Step 1
Eliminate $y$
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Answer
To find the points of intersection between the curve C and line L, we first express y from the equation of the line L:
y=5x−4.
Now, substitute this expression for y into the equation of the curve C:
5x−4=x2+3x−3.
Step 2
Rearranging and Collecting Terms
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Answer
Rearranging the equation gives us:
x2+3x−5x+4−3=0
which simplifies to x2−2x+1=0.
Step 3
Factoring and Solving
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Answer
Factoring the quadratic equation yields:
(x−1)2=0.
This means we have a repeated root at x=1.
Step 4
Statement Concluding One Point in Common
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Answer
Since there is exactly one solution for x (i.e., x=1), we find the corresponding y:
y=5(1)−4=1.
Thus, C and L have exactly one point in common at the coordinate (1,1).
