Exercises: Quadratic equations

Identify which of the following equations are quadratic. For those that are, write them in their 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) $(x + 2) \cdot (x - 2) = 0$

Calculate the solution to these incomplete quadratic equations of the type $ax^2 + c = 0$ (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 extracting 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) $4x^2 - 2x = 0$

We want to design a square rug for a room. If we know that the total area it must occupy is $144$ $m^2$, how long must the side of the rug be? Write the corresponding quadratic equation and solve it.

Without solving the equations, calculate the value of the discriminant ($\Delta = b^2 - 4ac$) and indicate 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) $2x^2 + 3x + 5 = 0$

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) $x^2 - 10x + 25 = 0$

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

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

A town council wants to install a square advertising banner on the facade of a sports center to announce an eSports event. If the total surface area occupied by the banner is $144 m^2$, set up a quadratic equation and calculate the side length of this banner.

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

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

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

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

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

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

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 feet. Solve the equation using the quadratic formula to find the possible values of $x$.

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

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

Write Emma's equation in its general form ($x^2 + bx + c = 0$). Hint: Use the factorized form $(x - x_1) \cdot (x - x_2) = 0$.

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

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

A parabolic ramp is going to be built for a new skatepark. For the ramp to be safe and have a single point of support on the ground, the discriminant of the equation describing its trajectory, $x^2 - 10x + k = 0$, must be equal to zero.

What value must the coefficient $k$ have for the ramp to have a single solution (repeated root)? Once $k$ is found, solve the equation to find the point of support.

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

If we call the height on the wall $x$, 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 one plus $10$. Set up the corresponding quadratic equation and find what these two numbers are. Is there more than one pair of numbers that satisfies the condition?

An urban artist wants to paint a square mural on a wall. However, they decide to enlarge the original design by increasing its side by $2$ feet. By doing so, the total area of the mural triples compared to the original design.

Calculate how long the side of the mural was originally (round the result to two decimal places if necessary).

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

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

On a photography social network, the number of "Likes" on an influencer's post today has been such that, if $10$ is subtracted from it and the result is squared, the same value is obtained as multiplying the number of "Likes" by $5$ and adding $50$.

Calculate how many "Likes" this post has received today.

A rectangular sheet of cardboard has a base of $20$ in. and a height of $15$ in. We want to cut out a square of side $x$ from each of its corners to make an open box. If we know that the base area of the resulting box must be $150$ square inches, calculate how long the side $x$ of the cut-out squares must be.