Exercises: Quadratic equation

Solve the following complete quadratic equation by applying the quadratic formula:

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

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

$3x^2 - 12x = 0$

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

$2x^2 - 18 = 0$

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

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

Write a quadratic equation in the form $ax^2 + bx + c = 0$ (with $a=1$) knowing that its solutions are $x_1 = 4$ and $x_2 = -3$.

Expand the expressions, simplify and solve the resulting quadratic equation:

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

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

Find the roots of the polynomial $P(x) = 2x^2 + 5x - 3$ and use them to write the factored polynomial.

An architect is designing a rectangular solar panel for the roof of an industrial building. Due to design constraints, the base of the panel must measure exactly $3$ feet more than its height. If the total area of the panel must be $28 \text{ ft}^2$, set up a quadratic equation and calculate its dimensions.

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

A technology company manufactures external hard drives. The monthly profit, in thousands of dollars, when selling $x$ hundreds of units is modeled by the quadratic function $B(x) = -2x^2 + 24x - 54$. How many units must be sold for the profit to be zero (break-even point)?

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

A photograph $20 \text{ cm}$ 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.

Solve the following equation with algebraic fractions by working it out until it is reduced to a quadratic equation:

$\frac{x+1}{x-2} + \frac{3x}{x+2} = 4$

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

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

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

Let $x_1$ and $x_2$ be the roots of the equation $3x^2 - 5x + 1 = 0$. Without explicitly calculating the value of $x_1$ and $x_2$, use the properties of the roots to find the exact value of the expression $\frac{1}{x_1^2} + \frac{1}{x_2^2}$.

In a right triangle with legs $6 \text{ cm}$ and $8 \text{ cm}$, a rectangle is inscribed so that two of its sides lie on 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.

Two surveillance drones depart simultaneously from the same base. The first drone flies in a straight line toward the north at a constant speed of $v \text{ km/h}$, and the second flies toward the east at a speed of $v + 5 \text{ km/h}$. After $2$ hours of flight, the straight-line distance separating them is $50 \text{ km}$. Set up a quadratic equation and determine the speed of each drone.

Prove algebraically that the quadratic equation $ax^2 + bx + c = 0$ (with $a \neq 0$) always has two distinct real solutions if the coefficients $a$ and $c$ have opposite signs (that is, if $a \cdot c < 0$ is satisfied).

Solve the following fourth-degree equation by making a suitable change of variable to transform it into a quadratic equation:

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

The golden ratio is obtained by dividing a segment of length $L$ into two parts $x$ and $y$ (with $x > y$) in such a way that the ratio of the total segment to the larger part is equal to the ratio of the larger part to the smaller part. If we set the length of the smaller part as $y = 1$, set up the resulting quadratic equation for $x$ and find the exact value of the golden ratio $\Phi$.

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