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 find the values of p and q, start by expanding the expression.

From the equation:

4x - 5 - x^3 = q - (x + p)^2

Expanding $(x + p)^2$ gives:

(x + p)^2 = x^2 + 2px + p^2

Substituting this back into the equation, we have:

4x - 5 - x^3 = q - (x^2 + 2px + p^2)

grouping like terms:

4x - 5 - x^3 = q - x^2 - 2px - p^2

Now, rearranging gives:

-x^3 + x^2 + (4 + 2p)x + (q + p^2 - 5) = 0

From this we can equate coefficients. For the coefficient of x^3 to be -1, we have:

The coefficient in front of x^2 is +1, thus q = 1.
For the coefficient of x, we see: $4 + 2p = 0$ , thus $p = -2$ .

Final answer: $p = -2$ , $q = 1$ .

Step 2

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

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Answer

To calculate the discriminant, we use the standard quadratic formula:

The general form of a quadratic equation is:

ax^2 + bx + c = 0

Here, we can rewrite the equation in standard form:

-x^3 + 4x - 5 = 0 ightarrow -x^2 + 4x - 5 = 0

Giving us coefficients:

$a = -1$
$b = 4$
$c = -5$

Now, the discriminant $riangle$ is given by the formula: $riangle = b^2 - 4ac$ Substituting the values gives: $riangle = 4^2 - 4(-1)(-5) = 16 - 20 = -4$ Thus the discriminant 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, start by determining where the curve crosses the axes:

Finding the x-intercepts: Set $y = 0$ :

0 = 4x - 5 - x^3 ightarrow x^3 - 4x + 5 = 0

This cubic equation can be solved for real roots (use numerical solutions or graphical analysis). 2. **Finding the y-intercept:** Set $x = 0$:

y = 4(0) - 5 - (0)^3 = -5

So, the y-intercept is (0, -5). 3. **Sketch the curve shape** by noting that as $x o - ext{\infty}$, $y o ext{\infty}$ (as a cubic has a positive leading coefficient). 4. **Maximum in the 4th quadrant:** The curve has zeros in the first and third quadrants, which can be shown with a sketch indicating a maximum The sketch should be made ensuring it accurately reflects the intercepts and overall shape.

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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