A curve has the equation y = x² + 4x - 4.
A straight line has the equation y = 3x - 2.
Work out the two points of intersection of the curve and the straight line.
How to approach this question
1. The points of intersection are where the (x, y) coordinates are the same for both equations. This means you can set the expressions for y equal to each other.
2. Set up the equation: x² + 4x - 4 = 3x - 2.
3. Rearrange this equation to form a quadratic equation in the standard form ax² + bx + c = 0.
4. Solve the quadratic equation for x. You can do this by factorising, completing the square, or using the quadratic formula.
5. You will get two different values for x.
6. For each value of x, substitute it back into one of the original equations (the linear one is easier) to find the corresponding y-value.
7. Write your final answers as two coordinate pairs (x, y).
Full Answer
To find the points of intersection between a curve and a line, we solve their equations simultaneously. Since both equations are given in the form "y = ...", we can set the right-hand sides equal to each other.
Step 1: Equate the equations.
x² + 4x - 4 = 3x - 2
Step 2: Rearrange into a standard quadratic form (ax² + bx + c = 0).
We need to move all terms to one side.
Subtract 3x from both sides:
x² + (4x - 3x) - 4 = -2
x² + x - 4 = -2
Add 2 to both sides:
x² + x - 4 + 2 = 0
x² + x - 2 = 0
Step 3: Solve the quadratic equation for x.
We can solve this by factorising. We are looking for two numbers that multiply to make -2 and add to make +1. The numbers are +2 and -1.
(x + 2)(x - 1) = 0
The solutions are found by setting each bracket to zero:
- If x + 2 = 0, then x = -2.
- If x - 1 = 0, then x = 1.
Step 4: Find the corresponding y-coordinates.
We substitute each x-value back into one of the original equations. The linear equation y = 3x - 2 is simpler to use.
- For x = -2:
y = 3(-2) - 2 = -6 - 2 = -8.
The first point of intersection is **(-2, -8)**.
- For x = 1:
y = 3(1) - 2 = 3 - 2 = 1.
The second point of intersection is **(1, 1)**.
Common mistakes
✗ Making an error when rearranging the equation to form the quadratic.
✗ Incorrectly factorising the quadratic equation.
✗ Finding the correct x-values but forgetting to find the corresponding y-values.
✗ Substituting the x-values back into the more complex quadratic equation, which increases the chance of an arithmetic error.
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