Exercises: Quadratic equations

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

$4x^2 - 64 = 0$

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

$3x^2 + 15x = 0$

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

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

Determine whether the equation $2x^2 + 3x + 5 = 0$ has solutions in the set of real numbers by first calculating the value of its discriminant $\Delta$.

Find the solutions to the following equation, which is already factorised, without expanding the product:

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

Write down a quadratic equation in the form $x^2 + bx + c = 0$ given that its only solutions are $x_1 = 3$ and $x_2 = -8$.

Find the value of the coefficient $c$ in the equation $x^2 + 4x + c = 0$ so that it has a single repeated real solution.

The square of a natural number minus three times the number is exactly equal to $18$. Set up a quadratic equation and find the number.

Lucas is solving the equation $2x^2 - 10x = 0$ on the blackboard and carries out the following steps in his notebook:

When checking the book's answer key, he is surprised to find that the equation actually has two solutions: $x_1 = 0$ and $x_2 = 5$.

1. a) Explain with mathematical precision what conceptual error Lucas made by dividing both sides by the unknown $x$ and why that operation eliminated one of the valid solutions.
2. b) Solve the equation correctly by factoring out the common factor and justify each step by applying the zero-product property.

We want to construct a quadratic equation with integer coefficients of the form $ax^2 + bx + c = 0$ whose exact solutions are $x_1 = -\frac{3}{2}$ and $x_2 = 4$.

1. a) Apply Cardano-Vieta's formulas to deduce the exact value of the sum ($S = x_1 + x_2$) and the product ($P = x_1 \cdot x_2$) of the solutions as an irreducible fraction.
2. b) Write an equivalent equation that has integer coefficients ($a, b, c \in \mathbb{Z}$), without denominators, with $a > 0$ and such that the coefficients share no common factors (that is, they are coprime).
3. c) Is the equation obtained in the previous section the only quadratic equation in the world that has exactly those two solutions? Justify your answer by reasoning what happens when multiplying all the coefficients by a constant $k \neq 0$.

To renovate the rectangular living room of a house, the architect states that the length is $2\text{ m}$ more than the width. If the total area of the living room is $35\text{ m}^2$, set up a quadratic equation and calculate the exact dimensions of the room.

An aerospace engineer launches a test rocket. The height $h$ (in metres) reached by the rocket as a function of time $t$ (in seconds) is given by the expression $h(t) = 40t - 5t^2$. Determine the times at which the rocket is at a height of exactly $75\text{ m}$.

Expand the algebraic identities, collect like terms and solve the following equation:

$(x + 2)^2 - 2(x - 1)^2 = 3x + 6$

Ticket sales for the school musical yield a profit $B$ (in tens of euros) that depends on the number of tickets sold $x$ (in tens) according to the equation $B(x) = -x^2 + 50x - 400$. Calculate exactly how many tickets must be sold for the profit to be zero (the break-even point).

Determine the numerical value of the parameter $k$ so that the equation $3x^2 - kx + 12 = 0$ has a single repeated real solution.

On a skateboard ramp in the shape of a right-angled triangle, the longer leg is $7\text{ m}$ longer than the shorter leg, and the hypotenuse is $2\text{ m}$ longer than the longer leg. Set up a quadratic equation using Pythagoras' theorem and calculate the lengths of the three sides of the ramp.

Solve the following equation by first operating with algebraic fractions. Make sure to check the solutions obtained to rule out any false roots that make the denominators zero:

$\frac{x}{x-2} + \frac{2x}{x+2} = \frac{16}{x^2-4}$

Given that one of the solutions to the equation $2x^2 + bx - 15 = 0$ is $x_1 = 3$, calculate without solving the equation completely the value of the other solution $x_2$ and the value of the coefficient $b$.

To solve the equation $(3x - 1)^2 - 25 = 0$, Alba and Mateo propose different strategies:

• Alba proposes expanding the binomial identity, subtracting $25$, collecting terms into the general standard form $ax^2 + bx + c = 0$, and applying the quadratic formula with the discriminant.

• Mateo proposes isolating the power directly, writing $(3x - 1)^2 = 25$, and taking the square root of both sides considering the double sign $\pm$.

1. a) Solve the equation following Alba's method step by step and then Mateo's, checking that both lead to the same solutions.
2. b) Compare both approaches and justify which one is more efficient and carries a lower risk of making calculation errors (signs, operations with large numbers).
3. c) Would Mateo's shortcut method be directly applicable to solve the equation $(3x - 1)^2 - 6x = 25$? Justify why or why not.

According to technical studies by the Directorate-General for Traffic (DGT), the total stopping distance $d$ (in meters) of a passenger car on dry asphalt as a function of the speed $v$ (in $\text{km/h}$) at which it is traveling is approximated by the quadratic model:

where $0,3v$ corresponds to the distance traveled during the driver's reaction time and $0,005v^2$ to the mechanical braking. The Local Police are investigating an accident on an urban avenue with a generic speed limit of $50\text{ km/h}$ and find a total stopping mark of $36\text{ m}$.

1. a) Estimate: Before solving the equation, calculate mentally or with quick trial and error what distance stopping would require at $50\text{ km/h}$ and at $70\text{ km/h}$. From these values, narrow down between which tens of $\text{km/h}$ the vehicle's speed must lie.
2. b) Set up the quadratic equation corresponding to $d = 36\text{ m}$ and solve it using the quadratic formula to obtain the exact numerical value of $v$.
3. c) Check: Rule out one of the mathematical solutions obtained using physical reasoning and determine whether the vehicle exceeded the legal speed limit, comparing the result with your previous estimate.

A rectangular photograph with base dimensions $20\text{ cm}$ by height $15\text{ cm}$ is to be framed with a mount of uniform width $x$. If the total area of the set (photograph plus mount) must be exactly $500\text{ cm}^2$, set up a quadratic equation and calculate the width $x$ of the mount.

In a volleyball league, each team plays exactly one match against every other team. Given that the total number of matches played in a tournament of $n$ teams is given by the formula $\frac{n(n-1)}{2}$, and that by the end of the season a total of $105$ matches have been played, set up the corresponding equation and find out how many teams took part in the league.

Two hoses of different thickness fill a water tank together in $6\text{ hours}$. If only one of them were used, the thinner hose would take $5\text{ hours}$ longer than the thicker hose to fill the tank on its own. Set up a quadratic equation by adding the fractions of the tank filled by each hose in one hour, and calculate how long each hose would take working individually.

Discuss, in terms of the different values of the parameter $m$, the number of real solutions to the following quadratic equation:

$mx^2 - 2(m-1)x + m = 0$

Make sure to analyse what happens when the discriminant is positive, zero or negative, and do not forget the special case where the leading coefficient is zero.

A sculptor wants to divide a brass rod of length $1\text{ metro}$ into two pieces of lengths $x$ and $1-x$ (where $x$ is the larger piece, that is, $x > 1-x$). They wish to do it in such a way that the ratio of the whole rod to the larger piece is equal to the ratio of the larger piece to the smaller piece. Set up the corresponding equation and find the exact value of $x$.

The area of a rectangle is $60\text{ cm}^2$ and the length of its diagonal is $13\text{ cm}$. Calling $x$ the base and $y$ the height, express $y$ in terms of $x$ using the given area. Then, substitute into Pythagoras' theorem for the diagonal and solve the resulting biquadratic equation to find the exact dimensions of the rectangle.

Solve the following equation containing a radical, first isolating the square root and squaring both sides. You must check the solutions obtained in the original equation to rule out any false roots:

$\sqrt{2x + 7} - x = 2$

A group of students hires a bus for a trip at a total cost of $400\text{ €}$. At the last minute, $5$ more students join the trip, which means that each of those who were originally going pays $4\text{ €}$ less. Set up a quadratic equation to calculate how many students were originally going, and solve it.

An olive-growing cooperative in Jaén dedicated to the production of Extra Virgin Olive Oil (EVOO) manages an estate with a standard density of $120$ olive trees per hectare, obtaining an average yield of $50\text{ kg}$ of olives per tree per year. Agricultural technicians warn that, for each additional olive tree $x$ planted per hectare, competition for soil water and nutrients reduces the average yield of each tree by $0,25\text{ kg}$.

1. a) Write the algebraic expression for the total production per hectare as a function of the number of added trees $x$, multiplying the total number of olive trees by the olive yield per tree.
2. b) The cooperative plans the harvest season to collect a total of $6\,375\text{ kg}$ of olives per hectare. Set up the corresponding quadratic equation, rearrange it into its standard form $ax^2 + bx + c = 0$, and calculate the two possible values of $x$.
3. c) The algebraic model provides two positive real solutions. Interpret the practical meaning of each of them and justify with reasoning which of the two options the cooperative should adopt if its goal is to minimize pruning, harvesting, and water consumption costs.

Consider the following family of equations involving a real parameter $k$:

1. a) A student claims that if $k = 2$, the equation completely lacks a solution because when applying the quadratic formula the denominator $2a$ vanishes ($2 \cdot 0 = 0$). Refute their claim with reasoning by explaining what type of equation is actually obtained when substituting $k = 2$ and calculate its solution.
2. b) For the cases where $k \neq 2$, calculate the simplified expression of the discriminant $\Delta$ as a function of $k$ and formally determine for which values of the parameter the equation has two distinct real solutions, one repeated real solution, or no real solutions.
3. c) Without the need to calculate the solutions of the equation, use Cardano-Vieta's relations in a justified manner to find out if there is any value of $k$ that makes the two roots symmetric or opposite ($x_1 + x_2 = 0$). If so, determine said value of $k$, check whether its roots are real, and find the value of both solutions.