Quadratic equations
You already solved quadratic equations last year. Now it is time to go beyond "applying the formula": understanding what the discriminant tells you about the nature of the solutions, extracting information without solving by using Cardano-Vieta, and discussing equations that depend on a parameter. This is the algebraic foundation you will need in Bachillerato.
General form and formula
Apply the quadratic formula to find the roots of the following complete quadratic equation: $x^2 - 7x + 10 = 0$
Apply the quadratic formula to find the roots of the following complete quadratic equation:
$x^2 - 7x + 10 = 0$
Find the solutions to the following equation, which is already factorised, without expanding the product: $(2x - 5)(x + 4) = 0$
Find the solutions to the following equation, which is already factorised, without expanding the product:
$(2x - 5)(x + 4) = 0$
Incomplete equations: solving without the quadratic formula
Solve the following incomplete quadratic equation by rearranging directly for the unknown: $4x^2 - 64 = 0$
Solve the following incomplete quadratic equation by rearranging directly for the unknown:
$4x^2 - 64 = 0$
Find the solutions to the following incomplete quadratic equation by taking out a common factor: $3x^2 + 15x = 0$
Find the solutions to the following incomplete quadratic equation by taking out a common factor:
$3x^2 + 15x = 0$
The discriminant: the nature of the solutions
Determine whether the equation $2x^2 + 3x + 5 = 0$ has solutions in the set of real numbers by first calculating the value of its discriminant $\Delta$.
Determine whether the equation $2x^2 + 3x + 5 = 0$ has solutions in the set of real numbers by first calculating the value of its discriminant $\Delta$.
Find the value of the coefficient $c$ in the equation $x^2 + 4x + c = 0$ so that it has a single repeated real solution.
Find the value of the coefficient $c$ in the equation $x^2 + 4x + c = 0$ so that it has a single repeated real solution.
Determine the numerical value of the parameter $k$ so that the equation $3x^2 - kx + 12 = 0$ has a single repeated real solution.
Determine the numerical value of the parameter $k$ so that the equation $3x^2 - kx + 12 = 0$ has a single repeated real solution.
Cardano-Vieta: sum and product without solving
This property also works in reverse: to build an equation from its solutions, without going through the product of binomials.
Write down a quadratic equation in the form $x^2 + bx + c = 0$ given that its only solutions are $x_1 = 3$ and $x_2 = -8$.
Write down a quadratic equation in the form $x^2 + bx + c = 0$ given that its only solutions are $x_1 = 3$ and $x_2 = -8$.
Given that one of the solutions to the equation $2x^2 + bx - 15 = 0$ is $x_1 = 3$, calculate without solving the equation completely the value of the other solution…
Given that one of the solutions to the equation $2x^2 + bx - 15 = 0$ is $x_1 = 3$, calculate without solving the equation completely the value of the other solution $x_2$ and the value of the coefficient $b$.
Equations with a parameter: discussing the number of solutions
In Bachillerato you will constantly meet equations with an extra letter (a parameter, usually $m$ or $k$) where you have to discuss, depending on its values, how many real solutions there are. The discriminant is once again the key tool, but with a new twist: if the coefficient $a$ depends on the parameter, you must also watch out for when $a=0$ (then the equation stops being quadratic).
Discuss, in terms of the different values of the parameter $m$, the number of real solutions to the following quadratic equation: $mx^2 - 2(m-1)x + m = 0$ Make sure to…
Discuss, in terms of the different values of the parameter $m$, the number of real solutions to the following quadratic equation:
$mx^2 - 2(m-1)x + m = 0$
Make sure to analyse what happens when the discriminant is positive, zero or negative, and do not forget the special case where the leading coefficient is zero.
Quadratic problems
To renovate the rectangular living room of a house, the architect states that the length is $2\text{ m}$ more than the width. If the total area of the living room is…
To renovate the rectangular living room of a house, the architect states that the length is $2\text{ m}$ more than the width. If the total area of the living room is $35\text{ m}^2$, set up a quadratic equation and calculate the exact dimensions of the room.
An aerospace engineer launches a test rocket. The height $h$ (in metres) reached by the rocket as a function of time $t$ (in seconds) is given by the expression…
An aerospace engineer launches a test rocket. The height $h$ (in metres) reached by the rocket as a function of time $t$ (in seconds) is given by the expression $h(t) = 40t - 5t^2$. Determine the times at which the rocket is at a height of exactly $75\text{ m}$.