Quadratic equations
You want a square rug of $144$ m² for your bedroom. How long must each side be? You are looking for a number $x$ such that $x^2 = 144$. This is a different kind of equation from the ones you already know: the unknown is raised to the power of two. These are called quadratic equations, and today you will learn to solve all of them, even the ones you cannot "see" at a glance.
General form
Graphically, $y = ax^2 + bx + c$ is a parabola. The solutions of the equation are the points where the parabola crosses the x-axis. Change $a$, $b$ and $c$ and see what happens:
Identify which of the following equations are quadratic. For those that are, write them in general form ($ax^2 + bx + c = 0$) and state the value of their coefficients…
Identify which of the following equations are quadratic. For those that are, write them in general form ($ax^2 + bx + c = 0$) and state the value of their coefficients $a$, $b$ and $c$:
- a) $3x^2 - 5 = x + 2$
- b) $x^2 + 3x - x^2 = 10$
- c) $5x - 3 = 0$
- d) $(x + 2) \cdot (x - 2) = 0$
Incomplete equations
When a term is missing, they can be solved without the formula, and more quickly:
Find the solution to these incomplete quadratic equations of the form $ax^2 + c = 0$ (remember that they can have two solutions, one, or none): a) $x^2 - 49 = 0$ b)…
Find the solution to these incomplete quadratic equations of the form $ax^2 + c = 0$ (remember that they can have two solutions, one, or none):
- a) $x^2 - 49 = 0$
- b) $2x^2 - 32 = 0$
- c) $x^2 + 25 = 0$
- d) $5x^2 = 0$
Solve the following incomplete quadratic equations by factorising out a common factor. Do not forget that one of the solutions is always $x = 0$: a) $x^2 - 7x = 0$ b)…
Solve the following incomplete quadratic equations by factorising out a common factor. Do not forget that one of the solutions is always $x = 0$:
- a) $x^2 - 7x = 0$
- b) $3x^2 + 12x = 0$
- c) $x^2 = 5x$
- d) $4x^2 - 2x = 0$
We want to design a square rug for a room. Given that the total area it must cover is $144$ $m^2$, what should the side length of the rug be? Write the corresponding…
We want to design a square rug for a room. Given that the total area it must cover is $144$ $m^2$, what should the side length of the rug be? Write the corresponding quadratic equation and solve it.
The quadratic formula
For a complete equation (with $b$ and $c$ both non-zero), there is a formula that always works:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$The symbol $\pm$ gives two solutions: one by adding the root and the other by subtracting it.
Without solving the equations, calculate the value of the discriminant ($\Delta = b^2 - 4ac$) and state whether they have two solutions, one repeated root, or no real…
Without solving the equations, calculate the value of the discriminant ($\Delta = b^2 - 4ac$) and state whether they have two solutions, one repeated root, or no real solutions:
- a) $x^2 - 6x + 9 = 0$
- b) $x^2 - 5x + 4 = 0$
- c) $2x^2 + 3x + 5 = 0$
A town council wants to install a square advertising banner on the facade of a sports centre to advertise an eSports event. If the total area covered by the banner is…
A town council wants to install a square advertising banner on the facade of a sports centre to advertise an eSports event. If the total area covered by the banner is $144 m^2$, set up a quadratic equation and calculate the side length of the banner.
The discriminant: how many solutions?
The expression inside the root, $\Delta = b^2 - 4ac$, is called the discriminant, and it tells you how many solutions the equation has without solving it:
| $\Delta = b^2 - 4ac$ | Solutions |
|---|---|
| $\Delta > 0$ | Two distinct solutions |
| $\Delta = 0$ | One solution (repeated) |
| $\Delta < 0$ | No real solutions |
Se va a construir una rampa parabólica para un nuevo skatepark. La trayectoria de la rampa es $y = x^2 - 10x + k$ y el suelo es el eje $x$ (es decir, $y = 0$). Para que…
Se va a construir una rampa parabólica para un nuevo skatepark. La trayectoria de la rampa es $y = x^2 - 10x + k$ y el suelo es el eje $x$ (es decir, $y = 0$). Para que la rampa sea segura y tenga un único punto de apoyo sobre el suelo, el discriminante de la ecuación $x^2 - 10x + k = 0$ debe ser igual a cero.
What value must the coefficient $k$ have for the ramp to have a single solution (a double root)? Once you have found $k$, solve the equation to find the support point.
Quadratic problems
When solving a problem, discard the solutions that make no sense: a negative length, a number of people with decimals…
In the mathematics lab, we are analysing the behaviour of different projectiles. Without solving the following equations, use the discriminant ($\Delta = b^2 - 4ac$) to…
In the mathematics lab, we are analysing the behaviour of different projectiles. Without solving the following equations, use the discriminant ($\Delta = b^2 - 4ac$) to state how many real solutions each one has:
- a) $x^2 - 6x + 9 = 0$
- b) $2x^2 + 3x + 5 = 0$
- c) $x^2 - 5x + 4 = 0$
The product of two consecutive integers is equal to four times the smaller number plus $10$ units. Set up the corresponding quadratic equation and find these two…
The product of two consecutive integers is equal to four times the smaller number plus $10$ units. Set up the corresponding quadratic equation and find these two numbers. Is there more than one pair of numbers that satisfies the condition?
Back to the start: For the $144$ m² rug, you solved $x^2 = 144 \Rightarrow x = \pm\sqrt{144} = \pm 12$; since a length must be positive, each side measures exactly $12$ m.