L3 Complex Numbers
Question Bank
Solve the equation $z^2 + 4z + 5 = 0$.
Express your solutions in the form $z = a \pm bi$, giving exact values for $a$ and $b$.
Solve the equation
$ 3x^2 - 4kx - 4k^2 = 0 $
for $x$ in terms of $k$, where $k$ is a real constant. Give your solutions in simplest form.
Solve the equation $z^2 + 8z + 41 = 0$.
Give your solutions in the form $a \pm bi$, where $a$ and $b$ are rational numbers.
For what value(s) of $k$ does the equation $x^2 + kx + 3k = 0$ have no real roots?
For what value(s) of $m$ does the equation $x^2 - mx + 4m = 0$ have no real roots?
Solve the equation $z^2 - 6z + 14 = 0$.
Express the solutions in the form $z = a \pm \sqrt{b}i$, where $a$ and $b$ are integers.
For what value(s) of $p$ does the equation $x^2 + \frac{6x}{p} + 1 = 0$ have equal roots?
Solve the equation $3z^2 - 4z + 6 = 0$.
Express your solutions in the simplest form $z = a \pm b\sqrt{c} i$, where $a, b, c$ are rational numbers.
Solve the equation
$ x(x + 2k) = 4k^2 $
for $x$ in terms of $k$, where $k$ is a real constant. Give your solutions in simplest form.
Find the value of $k$ if the roots of the equation $3x^2 - 12x + k = 0$ differ by 2.
(i.e. the roots are of the form $\alpha$ and $\alpha + 2$).
Find the equation whose roots are twice those of the equation $x^2 - 3x + 5 = 0$.
For what values of $m$, where $m$ is real, does the graph of $y = mx^2 - 2mx + 3$ not intersect the $x$-axis?
Prove that the equation $(px)^2 = 5 - \frac{2x}{p}$ has real roots for all real values of $p$, where $p \neq 0$.
Solve the equation
$ (z^2 + 2z)^2 - 2(z^2 + 2z) = 15 $
giving all four solutions in exact form. You must show algebraic working.
One root of the equation $x^2 + bx + c = 0$ is three times the other.
Find an expression for $c$ in terms of $b$.
One root of the equation $x^2 - kx + 18 = 0$ is twice the other.
Find the possible exact values of $k$.
Use the discriminant to find the equations of the tangents to the curve $y = x^2 + 2x + 2$ that pass through the point $(0, 1)$.
For each tangent, find the point where it touches the curve.
(a) Given that $z + \dfrac{1}{z} = p$, show that $z^2 + \dfrac{1}{z^2} = p^2 - 2$.
(b) Hence solve the equation
$ 2z^4 - 7z^3 + 9z^2 - 7z + 2 = 0 $
giving all four solutions in exact form.
Find the values of $k$ for which the equation $x + 2 - 3\sqrt{2x + k} = 0$ has no real roots.
Find the values of $k$ for which the equation $\sqrt{2x+k} - \sqrt{x} = 2$ has no real roots.