ExpertHard5 marksStructured
AQA GCSE · Question 24 · Algebra
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.
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
At the points of intersection, the y-values of the curve and the line are equal. So we can set the two equations equal to each other:
x² + 4x - 4 = 3x - 2
Now, we rearrange this into a standard quadratic equation (ax² + bx + c = 0):
Subtract 3x from both sides:
x² + x - 4 = -2
Add 2 to both sides:
x² + x - 2 = 0
Next, we solve this quadratic equation by factorising. We need two numbers that multiply to -2 and add to +1. These numbers are +2 and -1.
(x + 2)(x - 1) = 0
This gives two possible solutions for x:
x + 2 = 0 => x = -2
x - 1 = 0 => x = 1
Finally, we find the corresponding y-coordinate for each x-value by substituting them back into the linear equation (y = 3x - 2), as it's simpler.
- When x = -2:
y = 3(-2) - 2 = -6 - 2 = -8.
So one point is **(-2, -8)**.
- When x = 1:
y = 3(1) - 2 = 3 - 2 = 1.
So the other point is **(1, 1)**.
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.
Practice the full AQA GCSE Maths Higher Tier Paper 3 Calculator
32 questions · hints · full answers · grading
More questions from this exam
Q01Work out the reciprocal of 10/3. Give your answer as a decimal.EasyQ02.1The table shows information about the number of houses with solar panels in a town. Complete the ...EasyQ02.2Use the graph to estimate the number of houses with solar panels in 2023.EasyQ03.1A building in the shape of a cylinder has diameter 40 m and height 55 m. On the centimetre grid, ...MediumQ03.2On this centimetre grid, draw the front elevation of the building. Use a scale of 1 cm to 10 m.Medium