4x - 5 - x^3 = q - (x + p)^2 where p and q are integers. (a) Find the value of p and the value of q. (b) Calculate the discriminant of 4x - 5 - x^2. (c) On the axes on page 17, sketch the curve with equation y = 4x - 5 - x^3 showing clearly the coordinates of any points where the curve crosses the coordinate axes.

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4x - 5 - x^3 = q - (x + p)^2 where p and q are integers - Edexcel - A-Level Maths Pure - Question 9 - 2012 - Paper 2

Question 9

4x - 5 - x^3 = q - (x + p)^2 where p and q are integers. (a) Find the value of p and the value of q. (b) Calculate the discriminant of 4x - 5 - x^2. (c) On the ax... show full transcript

Worked Solution & Example Answer:4x - 5 - x^3 = q - (x + p)^2 where p and q are integers - Edexcel - A-Level Maths Pure - Question 9 - 2012 - Paper 2

Step 1

Find the value of p and the value of q.

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Answer

To start, we can equate coefficients from the two sides of the equation:

Setting the expanded form of $\text{q} - (x + p)^2 = - x^2 - 2px - p^2$ equal to $4x - 5 - x^3$ , we have:

4x - 5 - x^3 & = q - (x^2 + 2px + p^2) \ \text{Thus, compare constant terms:} \ -q & = -p^2 \ q = p^2 \ \text{For linear terms:} \ 4 & = -2p \ p & = -2 \ \text{So, substituting for p, we find:} \ \text{If } p = -2, \text{ then } q = (-2)^2 = 4. \text{The values are } p = -2 \text{ and } q = 4. \end{align*}$$

Step 2

Calculate the discriminant of 4x - 5 - x^2.

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Answer

The quadratic expression is represented as:

$b^2 - 4ac = (-4)^2 - 4(1)(-5)$

Upon simplifying:

data: b = -4, a = 1, c = -5: $16 - 20 = -4$

Thus, the discriminant of the polynomial is -4.

Step 3

On the axes on page 17, sketch the curve with equation y = 4x - 5 - x^3 showing clearly the coordinates of any points where the curve crosses the coordinate axes.

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Answer

To sketch the curve, we need to determine where it intersects the axes:

Step 1: Finding x-intercepts - Set y = 0: $0 = 4x - 5 - x^3$

This results in the cubic equation: $x^3 - 4x + 5 = 0$ Using numerical or graphical methods will show approximate x-intercepts.

Step 2: Finding y-intercept - Set x = 0: $y = 4(0) - 5 - (0)^3 = -5$

The coordinates where the curve crosses the coordinate axes are:

x-intercepts (values from cubic solution)
y-intercept at (0, -5).

Step 3: Sketch with correct shape - Ensure the curve reflects a maximum point in the 4th quadrant and properly represents crossings at (-5, 0) on the x-axis.

4x - 5 - x^3 = q - (x + p)^2 where p and q are integers - Edexcel - A-Level Maths Pure - Question 9 - 2012 - Paper 2

Worked Solution & Example Answer:4x - 5 - x^3 = q - (x + p)^2 where p and q are integers - Edexcel - A-Level Maths Pure - Question 9 - 2012 - Paper 2

Find the value of p and the value of q.

Calculate the discriminant of 4x - 5 - x^2.

On the axes on page 17, sketch the curve with equation y = 4x - 5 - x^3 showing clearly the coordinates of any points where the curve crosses the coordinate axes.

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