Quadratic Equations

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 $c$:

1. a) $3x^2 - 5 = x + 2$

1. b) $x^2 + 3x - x^2 = 10$

1. c) $5x - 3 = 0$

1. d) $(x + 2) \times (x - 2) = 0$

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):

1. a) $x^2 - 49 = 0$

1. b) $2x^2 - 32 = 0$

1. c) $x^2 + 25 = 0$

1. 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 $x = 0$:

1. a) $x^2 - 7x = 0$

1. b) $3x^2 + 12x = 0$

1. c) $x^2 = 5x$

1. d) $4x^2 - 2x = 0$

Solve the following incomplete quadratic equations using the most appropriate method (taking out a common factor or taking the direct square root):

1. a) $3x^2 - 27 = 0$
2. b) $x^2 + 5x = 0$
3. c) $2x^2 = 8x$

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.

Solve the following complete quadratic equations using the quadratic formula:

1. a) $x^2 - 5x + 6 = 0$

1. b) $x^2 + 2x - 8 = 0$

1. c) $x^2 - 10x + 25 = 0$

A garden designer wants to surround a circular fountain with a flower border. The equation relating the measurements of the design is $x(x - 8) = -15$, where $x$ is a measurement in meters. Solve the equation using the quadratic formula to find the possible values of $x$.

Reduce the following expression to general form $ax^2 + bx + c = 0$ and find its solutions by applying the formula:

A video game programmer is designing the formula to calculate a character's critical experience. After simplifying the code, the resulting equation is as follows:

Expand, reduce to general form $ax^2 + bx + c = 0$ and find the values of $x$ that make the formula work.

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:

1. a) $x^2 - 6x + 9 = 0$

1. b) $x^2 - 5x + 4 = 0$

1. c) $2x^2 + 3x + 5 = 0$

In the mathematics lab, we are analysing the behavior of different projectiles. Without solving the following equations, use the discriminant ($\Delta = b^2 - 4ac$) to state how many real solutions each one has:

1. a) $x^2 - 6x + 9 = 0$
2. b) $2x^2 + 3x + 5 = 0$
3. c) $x^2 - 5x + 4 = 0$

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.

Marta and Javier are playing "guess the equation". Marta says: "My equation has the solutions $x_1 = 3$ and $x_2 = -5$, and the coefficient of the $x^2$ term is $1$".

Write Marta's equation in general form ($x^2 + bx + c = 0$). Hint: Use the factorised form $(x - x_1) \times (x - x_2) = 0$.

The product of an integer and the next consecutive integer is equal to $30$. Translate this problem into a quadratic equation and find which numbers they could be (remember that there are two possible pairs of numbers).

Find the dimensions of the rectangle in the figure, given that its area is $15$ $cm^2$. To do this, set up a quadratic equation and solve it using the quadratic formula.

A municipality wants to install a square advertising banner on the facade of a sports center 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.

Find an integer such that if you add its own square to it, the result obtained is $42$. Hint: Remember to move all the terms to one side to set the equation equal to zero.

The base of a new tablet is rectangular. We know that its base is $4 cm$ more than its height and its total area is $60 cm^2$. Calculate the dimensions (base and height) of the tablet by solving the corresponding quadratic equation.

An electric scooter $130$ cm long is leaning against a wall. The distance from the foot of the wall to the base of the scooter is $70$ cm less than the height the scooter reaches on the wall.

If we call $x$ the height on the wall, apply Pythagoras' theorem to set up a quadratic equation and calculate this height.

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?

A street artist wants to paint a square mural on a wall. However, he decides to enlarge the original design by increasing its side length by $2$ meters. In doing so, the total area of the mural is three times that of the original design.

Calculate the original side length of the mural (round the result to two decimal places if necessary).

A small toy drone is launched vertically upward from the ground. The height $h$ (in meters) of the drone after a time $t$ (in seconds) is given by the expression:

At what times will the drone be at a height of exactly $25$ meters? Explain why you get two different solutions.

On a photo-sharing social network, the number of "Likes" on an influencer's post today was such that, if $10$ units are subtracted from it and the result is squared, the value obtained is the same as multiplying the number of "Likes" by $5$ and subtracting $50$ from it.

Calculate how many "Likes" that post received today.

A rectangular sheet of cardboard has a base of $20$ cm and a height of $15$ cm. We want to cut out a square of side length $x$ from each of its corners to make an open box. Given that the base area of the resulting box must be $150$ cm², calculate the length of the side $x$ of the cut-out squares.