Exercises: Quadratic equation

Solve the following complete quadratic equation using the quadratic formula:

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

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

$3x^2 - 12x = 0$

Solve the following incomplete quadratic equation by solving directly for 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:

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

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

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

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

A photograph measuring $20 \text{ cm}$ 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.

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

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

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

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

Solve the following quartic equation by applying a suitable substitution to transform it into a quadratic equation:

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

A group of friends decides to rent a sailboat for the weekend for a total cost of $7\,200$ pesos, which they agree to split equally. At the last minute, two of the friends cannot attend, forcing the rest of the group to pay $180$ pesos more each to cover the total cost. Set up an equation that reduces to a quadratic to determine how many friends were in the original group.