Exercises: Quadratic equations

Solve the following incomplete quadratic equation by rearranging directly for the unknown:

$3x^2 - 48 = 0$

Find the solutions to the following equation by taking out a common factor:

$5x^2 + 15x = 0$

Apply the quadratic formula to find the roots of the following quadratic equation:

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

Calculate the discriminant, $\Delta = b^2 - 4ac$, and state the number of real solutions the equation has:

$4x^2 - 12x + 9 = 0$

Write a quadratic equation in the form $x^2 + bx + c = 0$ knowing that its solutions are $x_1 = 4$ and $x_2 = -3$.

Find the solutions to the following equation mentally, as it is already factorised:

$(2x - 5)(x + 4) = 0$

Find a natural number such that its square minus three times itself is equal to $18$. Set up the corresponding equation and solve it.

Solve the following equation by first eliminating the denominators to convert it into an equation with integer coefficients:

$\frac{x^2}{2} - \frac{x}{3} = \frac{5}{6}$

In a Mathematics exam, Sergio has to solve the incomplete equation $3x^2 - 15x = 0$. His working on paper was as follows:

And he concludes: "The only solution to the equation is $x = 5$".

1. a) Explain with reasoning what conceptual error Sergio makes by dividing both sides by the unknown $x$, and what direct consequence this has on the solutions.
2. b) Solve the equation correctly by factoring out the common factor and justify, using the zero-product property ($A \cdot B = 0$), why there are actually two distinct solutions.

In a municipal sports centre in Sevilla, a new standard padel court is going to be built. Following the official regulations of the Spanish Padel Federation, the perimeter of the court must measure $60\text{ m}$ (so the sum of its length and its width is $30\text{ m}$) and the total playing area must be $200\text{ m}^2$.

1. a) If we call $x$ the width of the court in metres, express its length in terms of $x$ and set up the quadratic equation that determines the area of the court.
2. b) Solve the equation to obtain the possible values of $x$.
3. c) Explain the geometric meaning of obtaining two different algebraic solutions: are these two different courts, or do they represent the length and width of the same court?

Solve the following equation by first expanding the algebraic identities and collecting like terms:

$(x - 2)^2 + (x + 1)^2 = 5$

The area of a rectangular solar panel is $24$ m$^2$. Knowing that its length is $2$ metres longer than its width $x$, calculate the exact dimensions of the panel.

A model rocket is launched vertically upwards. Its height $h$ in metres after $t$ seconds is given by the equation $h = 40t - 5t^2$. Determine at what times the rocket will be at a height of exactly $75$ metres.

In the equation $2x^2 + kx - 15 = 0$, one of the solutions is known to be $x = 3$. Calculate the value of the parameter $k$ and find the other solution.

The sum of the squares of two consecutive odd integers is $130$. Set up a quadratic equation, solve it and find these numbers (state all possible pairs).

A group of friends hires a van for $120$ euros for a trip, sharing the cost equally. At the last minute, $2$ more friends sign up, reducing the amount each attendee pays by $3$ euros. Set up the corresponding equation and find how many friends actually go on the trip.

Solve the following equation by performing algebraic operations on the fractions to transform it into a quadratic polynomial equation:

$\frac{x}{x-1} + \frac{2}{x+1} = \frac{8}{x^2-1}$

Without solving the equation $3x^2 - 12x + c = 0$, find the value of the constant term $c$ knowing that the product of its roots is $3$. Also calculate the sum of these roots using Vieta's formulas.

To solve the equation $2(x - 3)^2 - 50 = 0$, two classmates propose different approaches:

• Elena suggests expanding the special product $(x - 3)^2$, multiplying by $2$, subtracting $50$, rearranging the complete equation into the form $ax^2 + bx + c = 0$, and applying the general quadratic formula.

• Darío proposes isolating the squared term $(x - 3)^2$ by dividing by $2$ and solving directly by taking the square root with a double sign ($\pm$) on both sides.

1. a) Solve the equation following the method proposed by Darío.
2. b) Compare both strategies: which of the two methods is more efficient and reduces the likelihood of making errors with signs or intermediate operations? Justify your answer.
3. c) In which case would it be impossible to apply Darío's procedure, making it essential to resort to the general form and the quadratic formula?

The Directorate-General for Traffic (DGT) models the total stopping distance $d$ (in metres) of a passenger car on dry asphalt as a function of its speed $v$ (in $\text{km/h}$) using the following quadratic expression:

Following an incident on a conventional road with a speed limit of $90\text{ km/h}$, the Traffic Civil Guard measured a skid and stopping mark of $70\text{ m}$.

1. a) Estimate: Without solving the complete equation, calculate the distance the car would require at $80\text{ km/h}$ and at $100\text{ km/h}$. From these two values, reason between which speeds the vehicle was travelling and indicate whether you suspect the driver was exceeding the legal limit of $90\text{ km/h}$.
2. b) Calculate: Substitute $d = 70$ into the formula and multiply all terms by $200$ to work with integer coefficients. Solve the resulting quadratic equation to calculate the exact speed $v$.
3. c) Check and assess: Contrast the result with your previous estimate, discard with justification the mathematical solution that makes no physical sense, and determine whether the driver committed a speeding offence.

Discuss the number of real solutions to the equation $x^2 - (m+2)x + 9 = 0$ depending on the values of the parameter $m$.

In a right-angled triangle, the hypotenuse is $2$ cm longer than the longer leg, and the longer leg is $2$ cm longer than the shorter leg. Set up a quadratic equation using Pythagoras' theorem and calculate the exact perimeter of the triangle.

Given the roots $x_1$ and $x_2$ of the general equation $ax^2 + bx + c = 0$ (with $c \neq 0$), prove algebraically that the sum of the reciprocals of the roots, $\frac{1}{x_1} + \frac{1}{x_2}$, is always equal to $-\frac{b}{c}$.

The perimeter of a rectangular plot is $68$ metres and its area is $240$ m$^2$. Use the relationship between the sum and product of the roots of a quadratic equation ($x^2 - Sx + P = 0$) to determine the dimensions of the plot.

A photograph measuring $20$ cm wide by $30$ cm high is framed with a mount of uniform width $x$. If the total area of the mount is exactly equal to the area of the photograph, set up a quadratic equation and find the width $x$ of the mount.

Write a quadratic equation whose roots are double the roots of the equation $x^2 - 5x + 6 = 0$. Solve it by applying the properties of the sum and product of the roots, without first calculating the roots of the original equation.

Simplify and solve the following rational equation, taking care with possible spurious solutions that make the denominators zero:

$\frac{x+1}{x-2} - \frac{x-1}{x+2} = \frac{12}{x^2-4} + 1$

A freight train travels at a constant speed. A high-speed train, travelling $50$ km/h faster, covers a distance of $300$ km in $1$ hour less than the freight train. Set up a quadratic equation to determine the speed of the freight train.

In the structural design of a pedestrian bridge, the lower arch is described by a quadratic equation of the form $x^2 + bx + c = 0$, where the variable $x$ represents the horizontal position in metres relative to a central marker. Two structural conditions are established for the ground supports ($x_1$ and $x_2$):

1. The total separation between both supports must be $40\text{ m}$ (that is, $x_2 - x_1 = 40$).

2. Due to the unevenness of the ground, the sum of the horizontal positions of the supports is $x_1 + x_2 = 20$.

1. a) Without first calculating the value of each support, determine the value of coefficient $b$ by applying Vieta's formulas ($x_1 + x_2 = -\frac{b}{a}$).
2. b) Set up and solve the system of two linear equations formed by the sum and the difference of the supports to find the exact positions $x_1$ and $x_2$.
3. c) Deduce the value of the constant term $c$ using the product of the solutions ($x_1 \cdot x_2 = \frac{c}{a}$), write the complete general equation, and prove, from the sign of its discriminant $\Delta = b^2 - 4ac$, that the existence of two real and distinct supports is guaranteed.

A youth theatre company in Valladolid offers its performances at a regular price of $12\text{ €}$ per ticket, recording an average attendance of $300$ people per performance. A market study by the cultural association reveals that, for every euro ($1\text{ €}$) they decide to increase the ticket price, $15$ fewer spectators will attend. The revenue per performance is given by the product:

1. a) If we call $x$ the amount in euros by which the initial price is increased:

• Write in terms of $x$ the expressions for the new price and the new number of spectators.

• Expand their product and verify that the total revenue is given by a quadratic expression.

1. b) To finance a tour around Castilla y León, the company needs to raise exactly $3780\text{ €}$ in one performance. Set up the corresponding equation, simplify it by dividing all its terms by $-15$, and solve it.
2. c) Interpret the two results obtained: which two different ticket prices allow them to achieve that exact same revenue target? Which of the two options would you advise choosing if the company's social purpose is to make theatre accessible to as many citizens as possible? Justify your choice.