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) $5x^2 = 2x$

The area of the square base of a video game storage box is $144\text{ cm}^2$. Calculate the length of the side of the base by setting up a quadratic equation.

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 $ax^2 + bx + c = 0$.

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.

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) $x^2 + x - 12 = 0$

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).

Solve the following equation by simplifying both sides first and using standard algebraic identities where necessary:

$\frac{(2x - 3)^2}{8} - \frac{x(x + 1)}{2} = \frac{x - 5}{4} - 1$

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:

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

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:

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?

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?

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.

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

A surveillance drone takes off and follows a trajectory whose height in meters 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 $53 \text{ meters}$.

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).

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?