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The Quadratic Formula

The quadratic formula is a fundamental formula used to find the roots of quadratic equations. Roots of quadratic polynomials are defined as the values of the variables for which the quadratic polynomial is equal to zero. For example, given the general form of a quadratic polynomial

the roots of the above are defined as the values of x for which the above polynomial is equal to zero, i.e.

Because they are degree 2, quadratic polynomials ALWAYS have 2 roots. These roots can be positive, negative or zero and either real or complex (imaginary).

The quadratic formula is given as:

where

and in the case where the discriminant is 0 (which means there is one root)

Although it is important to memorize the formula, understanding how and why it exists is better.

Deriving the Quadratic Formula

A more general form of algorithm for finding roots of quadratic equations by completing squares leads to the derivation of what in known as the Quadratic Formula. This is a general formula which can be used to solve for the roots of any quadratic equation.

Given a quadratic equation

then the roots of the equation can be found by completing the square as below:

This can be further simplified as follows

Putting everything under the under one denominator results in

The above equation is known as the Quadratic Formula.

From this derivation, we can generalize a few equalities based on the formula.

For all real numbers b and c,

For all real numbers b and c,

For all real numbers a, b, and c where a does not equal 0,

The Discriminant of the Quadratic Formula

The part of the quadratic formula under the radical sign is referred to as the discriminant. This is because this expression is what determines if the quadratic equation whose roots we're trying to find has real roots, imaginary (complex) roots, or has the same root repeated. is important because this expression is under the square root sign. Remember that the square root of a number greater than zero (a positive number) is a real number, the square root of zero is zero and the square root of a number less than zero (a negative number) is an imaginary or complex number. Thus the value of says a lot about the nature of the roots of the quadratic equation.

  1. If

    i.e.

    This quadratic equation is said to have one repeated root. For example:

    looking at only

    The above would indicate that the equation has one repeated root, and we already saw that it does indeed have one repeated root.

    The roots of the equation are given by x = {-2,-2} which is the same root repeated.

  2. If

    then is greater than zero and the quadratic equation whose roots we're finding is said to have real roots. For example, if asked to find the roots of the given quadratic equation

    looking at only

    and 1 is greater than zero, we can conclude that the quadratic equation has real roots, which is proved by finding the roots of the equation using the quadratic formula.

    and

    Therefore the roots of the equation are given by x={-2,-1} which are both real numbers.

  3. If

    is less than zero and the quadratic equation whose roots we're finding is said to have complex or imaginary roots. A complex or imaginary number is denoted by "i", e.g. 4i is an imaginary number 4.

Given the quadratic equation , look at only to find the roots.

which is a negative indicating that the roots of the quadratic equation are imaginary. Knowing this, the roots can be found as follows:

Substituting into the quadratic formula

Since we already know that is a negative number, we can find the roots by making the following adjustment to the quadratic formula:

notice the i in front of the radical sign indicates that the number is imaginary.

which results in

which gives the roots

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