III. Quadratic Equations 8. Applications of Quadratic Equations

Let be the solution of a quadratic equation . Find the value of for rational numbers and .

The solutions of a quadratic equation are . For two rational numbers , , find the value of .

When the solutions of the quadratic equation are , for two rational numbers and , find the value of .

Find the sum of the solutions of .

When the quadratic equation has a multiple root, find the value of constant .

The quadratic equation has a multiple root . Find the value of .

Find the range of constant for the quadratic equation to have two different solutions.

When the solutions of the quadratic equation are and , find the value of .

When the solutions of the quadratic equation are and , for two constants and , find two solutions of the equation .

When a quadratic equation with coefficient for was given, Francesca misread the coefficient of and had the solutions , , while Nathaniel misread the constant term and had the solution , . Find the correct quadratic equation.

When the sum of the solutions of is , and the multiplication of them is , find the value of .

When and are the solutions of , find the value of .

When and are the solutions of the quadratic equation , find the value of .

When the difference of two solutions of the quadratic equation is , find the value of constant .

When the solutions of the quadratic equation are and , find the quadratic equation whose solutions are and with the coefficient for term.

When is a solution of the quadratic equation , find the value of rational number .

When is a solution of the quadratic equation , find the value of for the rational numbers and .

The number of diagonals of an -sided polygon is . For a -sided polygon having diagonals, find the natural number .

There are three consecutive natural numbers. Given that the square of the largest number is less than twice the product of the other two numbers, find the largest of these three numbers.

There are two natural numbers whose difference is and whose product is . Find the smaller one.

There are two sisters whose age difference is . If the sum of squares of their ages is , how old is the younger one?

When launching a ball with an air pump, the height from the ground after seconds of the launching ball can be computed as . Find how many seconds it takes for the ball to reach the height of for the first time.

As shown in the following figure, a square with sides and was created by holding a point on a line segment with a length of . Find the length of when the sum of the areas of the two squares is . (note, )

As shown in the following figure, a road with a constant width was built in a rectangular field with a width of and a length of . If the area of ​​the field excluding the road is , find the width of the road.