Quadratics: Question 1

Worked solution

Step 1: Identify $a$, $b$ and $c$

Compare $2x^2 - 8x + c$ with the general form $ax^2 + bx + c$:

$a = 2, \qquad b = -8, \qquad c = c$

Step 2: Apply the condition for two distinct real roots

A quadratic $ax^2+bx+c=0$ has two distinct real roots exactly when its discriminant is positive:

$b^2 - 4ac > 0$

Step 3: Substitute and simplify

$(-8)^2 - 4(2)(c) > 0$

$64 - 8c > 0$

$64 > 8c$

$c < 8$

Why the other options are wrong

• B ($c>8$): this reverses the inequality. It is actually the condition for no real roots, since $64-8c<0$ there.
• C ($c<4$): comes from confusing the $2a$ used in the quadratic formula’s denominator with the discriminant’s $4ac$ term, mistakenly computing $4(2a)c = 4(4)c = 16c$ instead of the correct $4ac = 8c$: $64 - 16c > 0 \Rightarrow c < 4$.
• D ($c>4$): combines both errors above, the same wrong coefficient $16c$ and the reversed inequality: $64 - 16c < 0 \Rightarrow c > 4$.

Final answers

• $c < \boxed{8}$, option A.