The curve C has equation $y = x^2 + 3x - 3$ - Edexcel - GCSE Maths - Question 17 - 2021 - Paper 3
Question 17
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 17 - 2021 - Paper 3
Step 1
Eliminating $y$
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Answer
To find the intersection of the curve C and the line L, we start by expressing y from the line L's equation:
y=5x−4.
Now, we can substitute this expression for y into the equation of the curve C:
5x−4=x2+3x−3.
Step 2
Rearranging Terms
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Answer
Next, we rearrange the equation to combine like terms:
0=x2+3x−5x+4−3,
which simplifies to:
0=x2−2x+1.
Step 3
Factoring the Quadratic
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Answer
We can factor this quadratic equation:
0=(x−1)2.
Step 4
Finding $y$ value and Conclusion
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Answer
Setting the factor equal to zero gives:
x−1=0⟹x=1.
Now substituting x=1 back into the equation of line L:
y=5(1)−4=1.
Thus, the point of intersection is (1,1), indicating that C and L have exactly one point in common at this coordinate.
