Equations & Their Roots — practice questions

Solve $5(x - 3) = 2(x + 6)$.

Make $h$ the subject of the formula $A = 2\pi r^2 + 2\pi rh$, and hence find $h$ when $A = 200$ and $r = 4$, correct to 3 significant figures.

A formula states $\dfrac{1}{f} = \dfrac{1}{u} + \dfrac{1}{v}$. Make $v$ the subject, and explain what happens to $v$ as $u$ approaches $f$.

Solve $x^2 - 5x - 24 = 0$.

Solve $3x^2 - 8x + 2 = 0$, giving your answers in exact surd form.

A quadratic equation with integer coefficients has roots $x = \dfrac{2}{3}$ and $x = -5$. Find the equation in the form $ax^2 + bx + c = 0$ where $a$, $b$ and $c$ are integers with no common factor, and explain what the fractional root tells you about the leading coefficient.

Use the discriminant to determine the nature of the roots of $2x^2 - 5x + 4 = 0$.

Find the values of $k$ for which $x^2 + (k-3)x + 4 = 0$ has equal roots.

Find the range of values of $m$ for which the line $y = mx - 2$ does NOT intersect the parabola $y = x^2 + 3$. Give your answer exactly.