Given that $2x^2 - 12x + p \equiv q(x-r)^2 + 10$, find the constants $p$, $q$ and $r$.
$p = 28$, $q = 2$, $r = 3$
- Without finding the roots, find the number of real roots of the equation $-2x^2 + 7x + 3 = 0$.
- The equation $2x^2 + px + x + 8 = 0$ has equal roots. Find the possible values of $p$.
- Discriminant is $73$, two real roots
- $p = -9, 7$
Solve the simultaneous equations $$x^2 - 3y + 11 = 0$$ $$2x - y + 1 = 0$$
$x = 2, y = 5 \quad x = 4, y = 9$
The length of a rectangle is $10$ m more than its width. The perimeter is greater than $64$ m. The area is less than $299$ m$^2$. Determine the set of possible values for the width of the rectangle.
$11 < x < 13$ where $x$ is the width
Prove that the sum of a rational number and an irrational number is always irrational.
Use proof by contradiction $\dfrac{a}{b} + n = \dfrac{c}{d}$ and $n$ is rational.