Exercises: Quadratic equations

For each of the following quadratic equations, identify the coefficients $a$, $b$ and $c$ and indicate 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, if the proposed values of $x$ are a solution to the equality $x^2 - 4x + 3 = 0$:

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 result of the square root is not a real number, you must indicate it:

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 extracting a common factor:

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

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

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

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

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

If we add its triple to the square of an integer, we obtain zero as a result. What numbers could it be? Set up the equation and find the possible solutions.

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

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

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?

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

Set up a quadratic equation and solve it.

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

Hint: Remember that if the first number is $x$, its consecutive is expressed as $x + 1$.

Solve the following incomplete quadratic equations using the fastest method (extracting a common factor or directly isolating 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) $5x^2 = 2x$

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

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

Solve the following equation by first calculating to simplify the sides and using algebraic identities where necessary:

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

A landscaping 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 how much the width $x$ of the border must measure.

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 (repeated root).

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

Two 3D printers working together take $2$ hours to complete a complex model. If they worked separately, the older printer would take $3$ hours more than the new printer. How long would each one 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 triangle knowing that its sides are three consecutive integers.

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

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

We have a cable of length $40 \text{ cm}$ which 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 much does the side of each of these squares measure?