L2 Algebra
Question Bank
Solve the equation $2x^2 - 9x - 5 = 0$.
Write $x^2 + 6x + 2$ in the form $(x+p)^2 - q$.
A function is defined as $f(x) = x^2 - 6x + 5$. Express $f(x)$ in completed square form, i.e. $f(x) = (x+a)^2 + b$, where $a$ and $b$ are integers.
A quadratic equation has roots $x = 3$ and $x = -5$.
Write the equation in the form $x^2 + bx + c = 0$.
A quadratic equation, $ax^2 + bx + c = 0$, has solutions of $x = \frac{1}{2}$ and $x = -4$. Find the values of the integers $a$, $b$, and $c$.
Alice was exploring an ancient forest when she came across a wise, talking oak tree. Curious about its age, Alice asked how old it was.
The tree creaked and boomed in reply, "I am very old indeed! In 190 years' time, my age will be the square of my age 190 years ago!"
Determine the tree's current age.
Hint: $210 \times 171 = 35910$ and $171 + 210 = 381$.
Solve the following equation for $y$:
$ \frac{\displaystyle 45}{\displaystyle y + 4} = y - 4 $
Find the values of x for which $x(2x+3) = 4 - 4x$.
Ben is thinking of a number. He performs the following steps:
- He squares the number.
- He multiplies the result by 4.
- He subtracts 8 times the number he was first thinking of.
- He then subtracts 60.
His final answer is 0. What numbers could Ben be thinking of?
Prove that $4x^2 - 4x + 1 \ge 0$ for all values of $x$.
Solve the following equation. Give your answer in exact form.
$2x + 5 = \frac{4}{x}$John is solving the equation $(3x+1)(x-2) = 15$ by using the quadratic formula.
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$Give the values of the integers $a$, $b$, and $c$ and hence solve the equation. Give your answer in exact form.
A rectangular garden is being planned. The length of the garden is to be 5 metres more than its width. If the area of the garden is planned to be 36 square metres, what are the dimensions (length and width) of the garden?
Express the function $f(x) = 3x^2 - 18x + 29$ in the completed square form $f(x) = a(x+h)^2 + k$.
Express the function $g(x) = 4x^2 + 8x - 1$ in the form $g(x) = a(x+h)^2 + k$, and hence state the coordinates of the vertex of the parabola.
Find a quadratic equation with integer coefficients whose roots are $x = \frac{2}{3}$ and $x = -\frac{1}{2}$.
One root of the equation $x^2 + bx - 12 = 0$ is $x = 3$.
Find the value of $b$ and the other root.
Solve the following system of simultaneous equations:
$m^2 + n^2 = 17$ $m - n = 3$
Find the coordinates of the points where the line $y = 2x + 1$ meets the parabola $y = x^2 - 2x - 4$.
Make $x$ the subject of the formula $y = 4x^2 + 5x - k$.
Find the value of $k$ for which the equation $2x^2 + kx + x + k = 4$ has one root equal to zero.
A parabola crosses the $x$-axis at $x = -\frac{3}{2}$ and $x = 4$, and crosses the $y$-axis at $(0, -24)$.
Find the equation of the parabola in the form $y = ax^2 + bx + c$.
Solve the equation
$(x^2 - 3)^2 - 4(x^2 - 3) - 12 = 0$
The equation $(x-1) + 2\sqrt{x-1} - 15 = 0$ has only one real solution. Find the value of $x$.
Solve the equation $4x^4 - 25x^2 + 36 = 0$. You must show algebraic working.
Solve the equation $u^{\frac{2}{5}} - u^{\frac{1}{5}} - 6 = 0$.
Solve the equation $25^x - 6 \cdot 5^x + 5 = 0$.
Find the value(s) of $k$ for which the equation $x^2 - 5 = (k^2 - k - 6)x$ has roots whose sum is zero.
Find the value of $k$ for which the equation $2kx^2 + 6kx + k = x^2 - 4$ has roots where one is the reciprocal of the other.
Using an algebraic method, solve the equation $2e^x - 4e^{-x} = 7$ for real values of $x$.
Solve the following equation for $x$, giving any solutions as exact values.
$(\log_3 x)^2 + \log_3(x^2) = 8$The equation $x^2 - mx = x-m$ has roots where one root is three times the other root. Find the values of the parameter $m$.
Find the values of $k$ for which the equation $2x^2 + (k-1)x + 9 = 0$ has one root that is double the other.
Find the value of $k$ for which the difference between the roots of the equation $x^2 + 4x + (k+1) = 0$ is 2.
The parabolas $y = x^2 - 3$ and $y = 2x^2 - 7x + 7$ meet at the points A and B.
(a) Find the equation of the straight line that passes through A and B.
(b) The second parabola is moved up or down so that its equation becomes $y = 2x^2 - 7x + k$. Find the value of $k$ for which the two parabolas touch at exactly one point.
Solve the following equation. You must check your solutions for extraneous roots and give your final answer in exact form.
$2x-1 = \sqrt{3x+4}$