Exercises: 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) $(x + 2) \cdot (x - 2) = 0$

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) $4x^2 - 2x = 0$

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.

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) $2x^2 + 3x + 5 = 0$

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 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.

Carlos is solving the incomplete quadratic equation $2x^2 - 10x = 0$ on the blackboard and carries out the following steps:

1) Moves the term with $x$ to the right-hand side: $2x^2 = 10x$.

2) Divides both sides by $x$: $\dfrac{2x^2}{x} = \dfrac{10x}{x} \Rightarrow 2x = 10$.

3) Solves for the unknown: $x = \dfrac{10}{2} = 5$.

Carlos concludes that the only solution to the equation is $x = 5$.

1. a) Explain what algebraic error Carlos made by dividing by $x$ and what direct consequence it has on the solutions of the equation.
2. b) Solve the equation correctly by factoring out the common factor and write down all its real solutions.

Design two quadratic equations yourself that meet the following conditions:

1. a) An incomplete equation of the form $ax^2 + c = 0$ whose solutions are $x = 4$ and $x = -4$. Explain the reasoning you followed to choose the values of $a$ and $c$.
2. b) An incomplete equation of the form $ax^2 + bx = 0$ (with $a \neq 0$ and $b \neq 0$) in which one of the solutions is $x = 0$ and the other is $x = -3$. Show step-by-step how you determined its coefficients.

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.

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.

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:

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 (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) $2x^2 = 8x$

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 metres. 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:

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 $(x - x_1) \cdot (x - x_2) = 0$.

In a school garden of an Andalusian secondary school, they want to fence a rectangular plot to grow vegetables. The design establishes that the length must be $4\text{ m}$ longer than the width and that the total enclosed area must be exactly $60\text{ m}^2$.

1. a) Estimate: Before setting up formulas, test mentally with simple integers ($x = 5$, $x = 6$, $x = 7$) and explain between which two integer values the actual measure of the width must lie.
2. b) Set up and solve: Call $x$ the width of the plot, write the quadratic equation in its standard form ($ax^2 + bx + c = 0$) and solve it using the quadratic formula.
3. c) Check: Reasonably discard the mathematical solution that makes no physical sense, calculate the definitive dimensions (width and length), and check whether the result confirms your initial estimate.

Look closely at the following three quadratic equations:

1) $4x^2 - 36 = 0$

2) $3x^2 + 15x = 0$

3) $x^2 - 7x + 10 = 0$

Sofía states: "To avoid getting confused with so many different rules, I always directly apply the quadratic formula $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to all equations."

1. a) Compare Sofía's strategy with the specific methods for incomplete equations. Explain which alternative method is faster and reduces the risk of errors in equation 1) and which one in equation 2).
2. b) Solve the three equations applying the most convenient and efficient procedure in each case, justifying your choice for each.

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.

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.

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 $2$ metres. 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) at which the drone is located after a time $t$ (in seconds) is given by the expression:

The graph shows the drone's trajectory and the height of $25$ m marked with a dashed line.

At what moments will the drone be exactly $25$ meters high? Solve it algebraically and 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.

During the Fallas festivities in Valencia, a pyrotechnics company fires a firework rocket vertically from the ground. The height $h$ (in meters) of the projectile above the ground as a function of elapsed time $t$ (in seconds) follows the physical model:

1. a) Set up a quadratic equation to find at what moments in time the rocket is at exactly $50\text{ m}$ in height. Solve it and interpret physically why two different time values are obtained.
2. b) If the device suffered a malfunction and did not explode in the air, how many seconds would it take to crash into the ground from the moment of launch? Model the corresponding equation, solve it, and discard the invalid solution with mathematical and physical justification.
3. c) The safety manager claims that the rocket will reach $70\text{ m}$ in height. Set up the equation corresponding to this situation and, without solving it completely, use the discriminant ($\Delta = b^2 - 4ac$) to argue whether the manager's claim is physically possible.

Consider the family of quadratic equations $x^2 - 8x + k = 0$, where $k$ is a real number that we can modify.

1. a) Use the discriminant ($\Delta = b^2 - 4ac$) to deduce what exact value $k$ must have for the equation to have a single repeated real solution. What is the value of that solution?
2. b) Determine for which values of $k$ the equation has no real solutions. Interpret geometrically what happens to the graph of the parabola $y = x^2 - 8x + k$ relative to the X-axis in that case.
3. c) Suppose we are told that one of the solutions to the equation is $x = 10$. Without using the discriminant formula, substitute this datum to find the exact value of $k$ and then, with proper reasoning, calculate the other solution of the equation.