Given that f(x) = 2x² + 8x + 3 (a) find the value of the discriminant of f(x) - Edexcel - A-Level Maths Pure - Question 8 - 2014 - Paper 1

To express f(x) in the desired form, we start from:

$f(x) = 2x^2 + 8x + 3$

We can factor out 2 from the first two terms:

$f(x) = 2(x^2 + 4x) + 3$

Now, we complete the square for the expression in the parenthesis:

$x^2 + 4x = (x + 2)^2 - 4$

Substituting back gives:

$f(x) = 2((x + 2)^2 - 4) + 3$

Expanding this:

$f(x) = 2(x + 2)^2 - 8 + 3 = 2(x + 2)^2 - 5$

Thus, we have:

• p = 2, q = 2, r = -5.

To find the value of c where the line y = 4x + c is tangent to the curve y = f(x), we first find the derivative of f(x):

$f'(x) = \frac{d}{dx}(2x^2 + 8x + 3) = 4x + 8$

The slope of the line y = 4x + c is 4, so we set:

$4x + 8 = 4$

Solving for x gives:

$4x = -4 \Rightarrow x = -1$

Next, we find the corresponding y-value on the curve:

$f(-1) = 2(-1)^2 + 8(-1) + 3 = 2 - 8 + 3 = -3$

Now substituting x = -1 into the line equation to find c:

$y = 4(-1) + c = -4 + c$

Setting these equal gives:

c = 1$$ Thus, the value of c is 1.