Quadratic equations

For each of the following quadratic equations, identify the coefficients $a$, $b$ and $c$ and state whether it is a complete or incomplete equation:

1. a) $x^2 - 5x + 6 = 0$
2. b) $2x^2 - 32 = 0$
3. c) $x^2 + 8x = 0$
4. d) $3 - x^2 + 2x = 0$

Check, without solving the equation, whether the proposed values of $x$ are solutions to the equation $x^2 - 4x + 3 = 0$:

1. a) $x = 1$
2. b) $x = 0$
3. c) $x = 3$
4. d) $x = -1$

Solve the following incomplete quadratic equations of the type $ax^2 + c = 0$. Remember that if the square root does not yield a real number, you must state so:

1. a) $x^2 - 25 = 0$
2. b) $3x^2 - 12 = 0$
3. c) $x^2 + 16 = 0$
4. d) $2x^2 = 50$

Solve the following incomplete quadratic equations of the type $ax^2 + bx = 0$ by taking out a common factor:

1. a) $x^2 - 7x = 0$
2. b) $2x^2 + 10x = 0$
3. c) $3x^2 = 6x$
4. d) $x^2 + x = 0$

Solve the following incomplete quadratic equations using the fastest method (taking out the common factor or rearranging directly for the square root):

1. a) $4x^2 - 100 = 0$
2. b) $3x^2 + 15x = 0$
3. c) $\frac{x^2}{2} - 8 = 0$
4. d) $5x^2 = 2x$

Apply the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the solutions to the following complete equations:

1. a) $x^2 - 6x + 8 = 0$
2. b) $x^2 + x - 12 = 0$

Solve the following quadratic equation by first simplifying the algebraic expressions through expanding the algebraic identities:

Remember that to apply the quadratic formula, you must first reduce the equation to the form $ax^2 + bx + c = 0$.

A surveillance drone takes off and follows a trajectory whose height in metres relative to time in seconds is given by the expression $h(t) = -2t^2 + 20t + 5$. Calculate at what times the drone is at a height of exactly $53 \text{ metres}$.

Solve the following equation. To do this, find the lowest common multiple of the denominators to eliminate the unknowns from them, and check that the solutions obtained do not make any denominator zero:

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

Calculate the value of the discriminant $\Delta = b^2 - 4ac$ for each equation and state whether it has two real solutions, a repeated root, or no real solutions (you do not need to calculate the solutions):

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

Study the discriminant ($\Delta = b^2 - 4ac$) of the following equations and determine, without solving them, how many real solutions each one has. Justify your answer in each case:

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

What value should the constant term take in equation a) so that it suddenly has no solution?

Given the quadratic equation $x^2 - (k-2)x + k = 0$, determine what values the parameter $k$ must take so that the equation has a single real solution (a repeated root).

Consider the family of quadratic equations $(k - 1)x^2 - 4x + 2 = 0$, where $k$ is a constant real number.

1. a) Analyze what happens if $k = 1$. Reason whether the resulting equation is still quadratic and find its solution in that particular case.
2. b) Assuming that $k \neq 1$, calculate the expression for the discriminant $\Delta$ in terms of $k$. From it, deduce with reasoning for which values of $k$ the equation will have two distinct real solutions, for which value it will have a single double real solution, and for which values it will have no real solution.
3. c) Choose a specific value of $k$ that has no real solution according to your previous study, substitute it into the original equation, and try to solve it using the formula to see what arithmetic obstacle appears.

Biquadratic equations can be solved by using a substitution $t = x^2$. Use this technique to find the four real solutions of the following equation:

Construct, in each case, a quadratic equation that satisfies the required conditions (with all terms grouped on the left-hand side and equal to zero):

1. a) An incomplete equation with no linear term ($b = 0$) whose quadratic coefficient is $a = 2$ and whose solutions are $x = 3$ and $x = -3$.
2. b) An incomplete equation with no constant term ($c = 0$) whose quadratic coefficient is $a = 2$ and whose solutions are $x = 0$ and $x = -5$.

Check the validity of your equations in both cases by solving them using the specific method suitable for each type of incomplete equation.

Given the quadratic equation $x^2 + kx + 12 = 0$, calculate the value of the coefficient $k$ knowing that one of the solutions to the equation is $x = 3$. Once $k$ is found, find the other solution of the equation.

If we add three times an integer to its square, we get zero as a result. What numbers could these be? Set up the equation and find the possible solutions.

The product of two consecutive natural numbers is equal to their sum increased by $19$ units. What are these numbers?

A video game programmer is designing an interface for a new app. They need to create a rectangular button whose base is $5\text{ cm}$ longer than its height. If the total area the button must occupy on the screen is $84\text{ cm}^2$, what should the dimensions (base and height) of the button be?

Set up a quadratic equation and solve it.

We want to put a wooden frame of constant width $x$ around a rectangular photograph measuring $10\text{ cm}$ by $15\text{ cm}$. If we know that the total area (photo plus frame) must be $300\text{ cm}^2$, what should the width $x$ of the frame be?

Galway City Council is planning a rectangular plot for community urban gardens. The technical specifications determine that the length of the land must be $6\text{ m}$ longer than its width and that the total cultivable area must be $160\text{ m}^2$.

1. a) Assign the unknown $x$ to the width of the plot, write the algebraic expression for its area, and set up the quadratic equation in its standard form $ax^2 + bx + c = 0$.
2. b) Solve the equation to determine the dimensions (length and width) of the garden. Explain in detail the geometric interpretation of each of the two mathematical solutions obtained and why you discard or keep each of them.

A landscape design company wants to surround a rectangular swimming pool $8 \text{ m}$ long by $6 \text{ m}$ wide with an artificial grass border of uniform width $x$. If the total area of the enclosure (pool plus grass border) must be exactly $80 \text{ m}^2$, calculate the width $x$ of the border.

Two 3D printers working together take $2$ hours to complete a complex scale model. If they worked separately, the older printer would take $3$ hours longer than the new printer. How long would each printer take to make the model on its own? (Hint: remember that the work done in one hour is the reciprocal of the total time).

Calculate the dimensions of a right-angled triangle knowing that its sides are three consecutive integers.

A piece of wire of length $40 \text{ cm}$ is cut into two pieces to form two different squares. If the sum of the areas of the two squares is $58 \text{ cm}^2$, how long is the side of each of these squares?

During a local summer festival in a town in County Galway, a firework rocket is launched vertically. The height $h$ (in meters) above the ground after $t$ seconds is given by the formula $h = -5t^2 + 40t$. The firework's detonator is programmed to explode during its ascent phase upon reaching exactly $60\text{ m}$ in height.

1. a) Before calculating formally, estimate whether the rocket will take more or less than $3\text{ seconds}$ to reach that height. To support your estimation, evaluate mentally or through simple calculations the height of the rocket at $t = 1\text{ s}$ and at $t = 2\text{ s}$.
2. b) Set up the quadratic equation $h = 60$, reduce it to its standard form, and solve it using the quadratic formula. Justify the physical meaning of the two values of $t$ obtained, check whether your previous estimation was correct, and indicate at which exact second the programmed explosion occurs.