L2 Algebra
Question Bank
Find the discriminant of the quadratic equation $2x^2 + 7x - 4 = 0$.
Find the discriminant of the quadratic function $y = 3x^2 - 10x + 7$ and hence determine the number of $x$-intercepts the graph has.
Find the discriminant of the quadratic equation $5x^2 - 9x = -2$ and hence determine the nature of its roots.
For what value of $p$ does the graph of the function $f(x) = 3x^2 - 12x + p$ just touch the x-axis?
Find the range of values of $k$ for which the parabola $y = kx^2 + 2x - 5$ has no x-intercepts.
The equation $(x-5)(3x+1) = k+1$ has only one real solution. Find the value of $k$.
The discriminant, $\Delta$, can be used to determine the number and nature of the roots of a quadratic equation. Complete the table below by filling in the missing information.
| VALUE OF THE DISCRIMINANT ($\Delta$) | NUMBER OF REAL ROOTS/SOLUTIONS | NATURE OF THE ROOTS |
|---|---|---|
| $\Delta >0$ | ||
| $\Delta = 0$ | Real and Equal (Rational) | |
| Rational and Distinct | ||
| No real roots |
Without solving the equation, use the discriminant to show that
$3x^2 - 7x + 2 = 0$
has two rational roots.
Change one coefficient in the quadratic $y = x^2 + 6x + 8$ so that its graph just touches the x-axis.
A farmer has 40 m of fencing to build a rectangular pen.
Use the discriminant to show that the pen cannot have an area of 120 m².
Find the possible values of $k$ if real solutions exist for $x^2 - 3x + 4 = k(x - 1)$.
A parabola is given by the equation $y=kx^2$, where $k$ is a non-zero constant. A straight line is given by the equation $y=mx+c$, where $m$ and $c$ are non-zero constants. Use algebra to prove that for the line to be a tangent to the parabola, the constants must satisfy the condition $m^2+4kc=0$.
The equation
$\small (p-2)x^2 + (p+1)x + (p-2) = 0$
is a perfect square. Find the value(s) of $p$.
Show that the roots of the equation $\small x^2 + 2(k-2)x - (k^2 - 4k + 7) = 0$ can never be equal for any real value of $k$.
The line $y = -3x + k$ is a tangent to the circle $x^2 + y^2 + 6x - 4y = 0$. Find the possible value(s) of $k$.
Find the range of values for $k$ for which the line $y = x + k$ does not intersect the circle $x^2 + y^2 = 9$.
The equation $x^2 + 6x + k = 0$, where $k$ is a positive whole number less than 10, has rational roots.
Find all the possible values of $k$.
The line $y = 2x + k$ is a tangent to the circle $x^2 + y^2 - 10x = 0$. Find the possible value(s) of $k$.
A drone flies in a straight line over a hill. Its path is modelled by $y = 0.5x + 2$, where $x$ is the horizontal distance in metres and $y$ is the height in metres.
The shape of the hill is modelled by $y = -0.1x^2 + 2x - 1$.
(i) Use the discriminant to show that the drone's path meets the hill.
(ii) The pilot raises the path, keeping the same gradient, so that it becomes $y = 0.5x + c$. Find the values of $c$ for which the drone's path clears the hill.
Show that the curve with equation $x^2 - 3xy - 40 = 0$ and the line with equation $3x + y + k = 0$ meet for all values of the constant $k$.