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. - 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 $y = x^2 + 3x - 3$
The line L has equation $y - 5x + 4 = 0$
Show, algebraically, that C and L have exactly one point in common. - Edexcel - GCSE Maths - Question 17 - 2021 - Paper 3
Step 1
Eliminate $y$
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Answer
To find the points of intersection between the curve C and the line L, first rewrite the equation of the line in the form of y: y=5x−4
Now, substitute this expression for y into the equation of curve C: 5x−4=x2+3x−3
Step 2
Rearranging Terms
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Answer
Next, let's rearrange the equation to combine like terms: 0=x2+3x−5x+4−3
This simplifies to: 0=x2−2x+1
This can be rewritten as: x2−2x+1=0
Step 3
Factoring the Quadratic
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Answer
The quadratic equation can be factored as follows: (x−1)2=0
This implies that x has a double root at x=1.
Step 4
Finding $y$ and Conclusion
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Answer
Substituting x=1 back into the equation of the line L to find the corresponding y-coordinate gives: y=5(1)−4=1
Hence, the point of intersection is (1,1). Since there is exactly one value of x and corresponding y, we conclude that the curve C and line L have exactly one point in common.
