Quadratic equations

Solve the following complete quadratic equation using the quadratic formula:

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

Given the following three quadratic equations:

(A) $3x^2 - 75 = 0$

(B) $2x^2 + 8x = 0$

(C) $x^2 - 5x + 6 = 0$

A classmate claims that the safest and fastest strategy consists of systematically applying the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to all three equally.

1. a) For each of the equations, compare the use of the quadratic formula with the most direct specific method (isolating the square root, factoring out a common factor, or mental calculation using Vieta's formulas). Justify with sound reasoning which method minimizes the risk of computational errors in each case.
2. b) Solve each of the equations by applying the optimal procedure you justified.

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

$3x^2 - 12x = 0$

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

$2x^2 - 18 = 0$

Cian has to solve the quadratic equation $5x^2 - 15x = 0$ on an exam. On his paper, he writes the following steps:

1) Rearrange the term: $5x^2 = 15x$.

2) Divide both sides by $x$: $\frac{5x^2}{x} = \frac{15x}{x} \implies 5x = 15$.

3) Solve for the unknown: $x = \frac{15}{5} = 3$.

Cian's conclusion: "The only solution to the equation is $x = 3$".

1. a) Identify and explain with algebraic precision the error Cian made in step 2 and its direct impact on the number of solutions found.
2. b) Solve the equation correctly by factoring out the common factor and justify why the zero-product property ($A \times B = 0 \iff A = 0 \text{ or } B = 0$) ensures that no solution is lost.

Without solving the equation, determine the number of real solutions it has by calculating the value of its discriminant:

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

Determine the value (or values) of the parameter $k$ such that the equation $x^2 - kx + 9 = 0$ has a single repeated real solution.

Given the equation $4x^2 - 5x - 9 = 0$, calculate the sum and product of its roots using Vieta's formulas, without needing to solve the equation.

Consider the quadratic equation $x^2 - 19x + 84 = 0$.

1. a) Without calculating the discriminant or applying the quadratic formula, use Vieta's formulas ($x_1 + x_2 = 19$, $x_1 \times x_2 = 84$) to estimate and bound with sound reasoning between which values the solutions must lie. First deduce the sign of both roots and test pairs of integers located at symmetric distances from half of the sum ($9.5$) until proposing a well-founded estimate of the solutions.
2. b) Solve the equation formally with the quadratic formula and compare the result obtained with your initial estimate. Explain why prior knowledge of the sum and product of the roots makes it possible to immediately detect potential computational errors during the resolution.

The goal is to determine a monic quadratic equation ($x^2 + bx + c = 0$, with $b, c \in \mathbb{R}$) whose unknown solutions $x_1$ and $x_2$ simultaneously satisfy two algebraic conditions:

1) The distance between both roots on the real line is equal to $4$ ($|x_1 - x_2| = 4$).

2) The sum of their reciprocals is $\frac{1}{x_1} + \frac{1}{x_2} = \frac{2}{3}$.

1. a) Relying on Vieta's formulas and using the algebraic identity $(x_1 - x_2)^2 = (x_1 + x_2)^2 - 4x_1 \times x_2$, set up a system that directly relates the coefficients $b$ and $c$ without calculating the individual roots beforehand.
2. b) Solve the system to deduce how many different quadratic equations simultaneously satisfy both requirements. Write down each of the resulting equations, find their respective solutions, and check that they satisfy the original conditions.

In the parametric equation $2x^2 + mx + 15 = 0$, one of the solutions is known to be $x = 3$. Without using the quadratic formula, calculate the value of the parameter $m$ and find the other solution.

Discuss the number of real solutions of the equation $(m-1)x^2 - 2mx + (m+2) = 0$ depending on the different values that the parameter $m$ can take.

Solve the following quartic equation by applying a suitable substitution to transform it into a quadratic equation:

$(x^2 - 3x)^2 - 2(x^2 - 3x) - 8 = 0$

In a physics experiment, a water rocket is launched vertically upwards. Its height $h$ in metres above the ground, as a function of time $t$ in seconds, is given by the expression $h(t) = -5t^2 + 20t + 2$. Determine the time or times at which the rocket is at a height of exactly $17$ metres.

A photograph measuring $20 \text{ cm}$ high by $30 \text{ cm}$ wide is to be framed with a wooden frame of uniform width $x$. If the total area of the frame (excluding the photograph) is $216 \text{ cm}^2$, set up an equation and find the width of the frame.

In a right-angled triangle with legs $6 \text{ cm}$ and $8 \text{ cm}$, a rectangle is inscribed such that two of its sides lie along the legs and one of its vertices lies on the hypotenuse. If the area of the rectangle must be exactly $12 \text{ cm}^2$, set up a quadratic equation and find the possible dimensions of the rectangle.

The product of two consecutive odd integers is $323$. Set up a quadratic equation and solve it to find these two numbers. Is there more than one possible pair?

A group of friends decides to hire a sailboat for the weekend for a total cost of $360$ euros, which they agree to split equally. At the last minute, two of the friends cannot attend, forcing the rest of the group to pay $9$ euros more each to cover the total cost. Set up an equation that reduces to a quadratic to determine how many friends were in the original group.

An Irish Rail train covers the direct route between Dublin and Cork, separated by a rail distance of $390\text{ km}$. Under normal conditions, the train travels at a constant average speed $v$ (expressed in $\text{km/h}$). Due to a temporary speed restriction imposed by maintenance work on the infrastructure, on a given day the train reduces its average speed by $65\text{ km/h}$, causing a delay of exactly $30\text{ minutes}$ ($0.5\text{ h}$) compared to its scheduled timetable.

1. a) Model the situation by relating distance, speeds, and travel times through an algebraic equation with unknown $v$, and transform it step by step into a quadratic equation of the form $a v^2 + b v + c = 0$.
2. b) Solve the obtained equation to find the regular speed $v$ of the train. Justify with a rigorous physical argument why one of the two mathematical solutions to the problem is discarded.
3. c) Calculate the time the train takes under normal conditions and the time taken on the day of the temporary restriction, verifying that the difference matches the data in the statement.