quadratic equations

Question

Why are there usually two solutions in quadratic equations? 5. can you please Give me an example from real life where quadratics are utilized. Why did you pick this example

Answer 1

Example:

You throw a rock straight up into the air with initial velocity of 7 m/s.

At what time does it reach the height of 2 m?

I picked this example because it illustrates what it takes to get two solutions,

one solution, or no solution.

I do not know whether you already took physics, and know how to solve this problem.

Regardless, let me just tell you.

While the initial velocity propels the rock upward, the rock is subject to

downward gravitational force; that force causes downward acceleration, called

g = 10 m/s² (approximately).

The upshot is that at any given time t, the rock reaches height

s(t) = 7t - gt²/2 = 7t - 5t².

To find the time when the rock reaches the height of 2 m, we write the equation

7t - 5t² = 2

or equivalently

5t² -7t + 2 = 0

You probably know that by the quadratic formula

t = (7 ± √(7² - 4×5×2))/2×5 = (7 ± 3)/10

That gives two solutions

t = 0.4 and t = 1

It is natural to get two solutions because the rock reaches the height of 2m

on its way up at time 0.4s, and then again on the way down at time 1s.

If instead we ask when the rock reaches the height of 2.45 m, we would get

t = (7 ± √(7² - 4×5×2.45))/2×5 = (7 ± 0)/10 = 0.7

We get only one solution because 2.45m is the maximal height, reached after 0.7s.

And if we asked when the rock reaches 3m, the answer would be "never";

the discriminant would be negative, giving us no real solution.

You asked why there are usually two solutions.

I assume you are wondering why two solutions, rather than just one.

There are a couple of answers:

Answer 1: The graph of a quadratic function is a parabola, and that tends to intersect the

x-axis in two points; but also sometimes just one, or none.

Answer 2: A quadratic function f(x) = ax² + bx + c

can be rewritten in the form f(x) = (x - r)(x - s).

It must be a product of two factors (x - r) and (x - s), not just one or three,

so as to get a term ax² after multiplication.

Given

(x - r)(x - s) = 0

This can be 0 only if at least of the factors is 0.

That means x - r = 0, or x - s = 0.

This gives two solutions.

Answer 2

As with typical x-y graphs, y changes as a function of x. In the case of a quadratic, the shape of the graph is a parabola. A solution, typically means the y-value must be a certain number. Here is an example:

Can you see that both the values x₁ and x₂ will result in a y-value denoted by the horizontal dotted blue line. So, the fact that there are often 2 solutions is a result of the shape of the graph of a quadratic.

A real life example could be the shape of the flight of a football as it is kicked down the field. "y" is the height, and "x" is the distance the ball travels. I'll let you decide why you picked this example (if you choose to use it).