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Quadratic Equations

 

 A quadratic equation is an equation with unknown variable to the second power.

The standard form of a quadratic equation is

 

 

 

 Where a, b, and c are constants and x is the unknown variable.

If an equation is not in standard form, we must manipulate it until it is.

Whenever we set y = 0 in any function, we are finding the x–intercept(s) for that function.

So when we solve a quadratic in the form we are really finding the x–intercepts.

These x–intercepts are also called roots.

Another term for roots or x-intercepts is zeros.

So, when we solve a quadratic in the standard form we are finding the roots of the function.

Roots and zeros are very important in electronics and control systems.

 

Solving Quadratic Equations

 

There are a variety of techniques for solving quadratic equations including:

 Factorization; Using the quadratic formula; Completing the square; Using a graph.

 We will discuss the first two techniques only.

 

Solving by Factorization

 

A number is said to be factorized when it is written as a product.

For example 24 can be factorized into 8 ´ 3. Algebraic expressions can also be factorized, as given

 

 

 

 

Example: Factorize z2 – 5z.

 

Solution

 

 

 

 

Solving Using a Formula

 

 Not all quadratics can be factorized. Accordingly, we will present the quadratic formula for solving the above equation. The quadratic formula is

 

 

 

 

Consider the following quadratic equation

 

 

 

Values of constants are: a = 1, b = 5, and c = 6.

These values can be substituted into the quadratic formula to give

 

 

 

 

There are two values of x:

the first one corresponds to the + sign and the second one correspond to the – sign. The answer is

 

 

 

Both values satisfy the original equation.

 

Example: The bending moment of a beam M, is given by the equation

 

 

 

 where x is the distance (m) along a beam from one end. Find the value of x for which M = 0.

 

Solution: We have

 

 

 We use to determine x.

Values of constants are: a = 7, b = 4, and c = -3, which can be substituted into the quadratic equation to give

 

 

 

 

 There are two values of x:

the first one corresponds to the + sign and the second one correspond to the – sign.

The answer is

 

 

 

Since we cannot have a distance of -1.00 on the beam, the bending moment M = 0 is at x = 0.429 m.