Linear and quadratic equation appears in several application problems. You can see the following problems as an application of quadratic equations.

1. A ball is thrown straight up, from 3 m above the ground, with a velocity of 14 m/s. When does it hit the ground? (Hint: ignore air resistance and take acceleration due to gravity equal to 9.8 m/s² approximately 10 m/s²)

Solution

Let t be the time required in seconds. Then the height h of the ball is given by,

h(t) = 3 + 14t -5t²

Observe that the height starts at 3 m, it travels at 14 m/s and gravity pulls it down, changing its position by about 5 m/s².

The ball hits the ground when the height is zero. Hence, solving 3 + 14t -5t² = 0, you get

t = -0.2 or t = 3

Since negative time is impossible, the time required is t = 3 s.
Therefore, the ball hits the ground after 3 seconds!

2. A company is going to make frames as part of a new product they are launching. The frame will be cut out of a piece of steel, and to keep the weight down, the final area should be 28 cm². The inside of the frame has
to be 11 cm by 6 cm. What should the width x of the metal be?

Solution

Consider the following as small steel frame:
Area of steel before cutting is given by,
Area = (11 + 2x) × (6 + 2x) cm²
Area = 66 + 22x + 12x + 4x²
Area = 4x² + 34x + 66

Finding the range of values of x for which y is increasing or decreasing/ Finding the range of values of x for which y is positive or negative

Domain: The function f(x) = x² + 5 is a quadratic function. It is defined for all values of x since there is no restriction on the value of x. Therefore, its domain is “all real values of x”.

Range: Since x² is never negative, the function is never less than 5. Therefore, its range is “all real numbers greater than or equal to 5.”

Domain: The function is a quadratic function. It is defined for all values of x since there is no restriction on the value of x. Therefore, its domain is “all real values of x”.

Range: Since is negative for each real value of x, the function is never greater than 9. Therefore, its range is “all real numbers less than or equal to 9.”