Exercises: Quadratic equations

Solve the following incomplete quadratic equation by isolating the unknown directly:

$4x^2 - 64 = 0$

Calculate the solutions of the following incomplete quadratic equation by factoring out the 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 a solution in the set of real numbers by first calculating the value of its discriminant $\Delta$.

Find the solutions of the following equation which is already factored, without needing to expand the product:

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

Write a quadratic equation in the form $x^2 + bx + c = 0$ knowing 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 calculate what the number is.

To renovate the rectangular living room of a house, the architect indicates that the length measures $2\text{ m}$ 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 feet) reached by the rocket in terms of the time $t$ (in seconds) is given by the expression $h(t) = 40t - 5t^2$. Determine at what times the rocket is exactly at a height of $75\text{ ft}$.

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

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

A company that manufactures electric scooters has determined that its daily profits $B$ (in hundreds of dollars) depend on the number of scooters manufactured $x$ according to the equation $B(x) = -x^2 + 50x - 400$. Calculate exactly how many scooters they must manufacture for the profit to be zero (the company's break-even point).

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

In a skating ramp shaped like a right triangle, the longer leg measures $7\text{ m}$ more than the shorter leg, and the hypotenuse measures $2\text{ m}$ more than the longer leg. Set up a quadratic equation using the Pythagorean theorem and calculate the length of the three sides of the ramp.

Solve the following equation by first working with algebraic fractions. Make sure to check the solutions obtained to rule out possible extraneous solutions that make the denominators zero:

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

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

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

In a volleyball league, each team plays exactly one game against all the others. Knowing that the total number of games played in a tournament of $n$ teams is given by the formula $\frac{n(n-1)}{2}$, and that at the end of the season a total of $105$ games have been played, set up the corresponding equation and find out how many teams participated in the league.

Two hoses of different thicknesses 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 thick hose to fill the tank on its own. Set up a quadratic equation by adding the fractions of the tank that each hose fills 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 that the following quadratic equation has:

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

Make sure to analyze what happens when the discriminant is positive, zero, or negative, and do not forget the special case in which 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 this 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 the base $x$ and the height $y$, express $y$ in terms of $x$ using the area data. Next, substitute into the Pythagorean theorem for the diagonal and solve the resulting biquadratic equation to find the exact dimensions of the rectangle.

Solve the following equation in which a radical appears, by first isolating the root and squaring both sides. It is mandatory to check the solutions obtained in the original equation to rule out possible extraneous solutions:

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

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