# Solution of Quadratic Equations

After reading this chapter, you should be able to:

1.find the solutions of quadratic equations,

2.derive the formula for the solution of quadratic equations,

3. solve simple physical problems involving quadratic equations.

** What are quadratic equations and how do we solve them?**

A quadratic equation has the form

The solution to the above quadratic equation is given by

So the equation has two roots , and depending on the value of the discriminant
,
, the equation may have real , complex or
repeated roots.

If
,
the roots are complex .

If
,
the roots are real .

If
,**
**the roots are real and repeated.

**Example 1**

Derive the solution to

**Solution**

Dividing both sides by a, , we get

Note if a=0, the solution to

is

Rewrite

as

**Example 2**

A ball is thrown down at 50 mph from the top of a building. The building is 420
feet tall. Derive the equation that would let you find the time the ball takes
to reach the ground.

**Solution**

The distance covered by the ball is given by

where

u= initial velocity (ft/s)

g= acceleration due to gravity (ft/s^{2})

t = time (s)

Given

we have

The above equation is a quadratic equation , the solution of which would give the time it would take the ball to reach the ground. The solution of the quadratic equation is

Since t > 0 the valid value of time is .

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