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

To express f(x) = 2x² + 8x + 3 in the required form, we start by factorizing the leading coefficient:

$f(x) = 2(x² + 4x) + 3$
Next, we complete the square inside the parentheses. To complete the square for x² + 4x, we take half of the coefficient of x, which is 4, resulting in 2, and then square it:
$2^2 = 4$
Now we insert this value into our equation and adjust for its addition:

$= 2((x + 2)² - 4) + 3$
$= 2(x + 2)² - 8 + 3$
$= 2(x + 2)² - 5$
Thus, we can identify p = 2, q = 2, and r = -5.

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

f'(x) = rac{d}{dx}(2x² + 8x + 3) = 4x + 8
Set the slope equal to the slope of the line at the point of tangency:
$4x + 8 = 4$
Solving for x gives:

ightarrow x = -1$$ Substituting x = -1 back into f(x): $$f(-1) = 2(-1)² + 8(-1) + 3 = 2(1) - 8 + 3 = -3$$ The point of tangency is (-1, -3). Substituting this point into the line equation to find c: $$-3 = 4(-1) + c ightarrow -3 = -4 + c$$ Thus, $$c = -3 + 4 = 1$$ Therefore, the value of c is 1.