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Welcome to our Math lesson on Quadratic Formula Definition, this is the first lesson of our suite of math lessons covering the topic of The Quadratic Formula, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
From tutorials 6.4 and 9.5, it is known that the general form of a quadratic equation is
where a and b are the coefficients (numbers followed by variables) of this equation while c is a constant (a number alone, with no variable after it).
In the previous tutorial, we explained a nice method for solving quadratic equations, that consists of completing the square. According to this method, we have to represent the expression in brackets in the form
Now, let's try to express the general form of a quadratic equation in such a way that fits the completing - the - square form. We have:
(This is because
)
From the last equation, we can write
or
or
Dividing both sides by a yields
Taking the square root of both sides in the last equation yields
or
or
In this way, we obtain two possible solutions for a quadratic equation:
and
The mathematical sentence
is known as the quadratic formula that allows us to solve any quadratic equation or detect immediately when a quadratic equation cannot be solved.
Solve the equation
using the quadratic formula and eventually check your solution by completing the square. Which method is simpler?
All quadratic equations have a general form of
Since our equation is
we have a = 2, b = 5 and c = 3.
Using the quadratic formula yields
and
Now, let's check the solution by completing the square. But first, we have to express the original equation in the form
We have to divide all terms of the original equation by 2 therefore. Thus, from
we obtain
Now, we have new coefficients for our equation: a = 1, b = 5/2 and c = 3/2. Let's write the expression in brackets that helps complete the square. Thus,
Since the simplified original equation is
or
we obtain
Thus,
and
As you see, the two roots are the same as those found by using the quadratic formula, but obtained through a longer procedure. Therefore, the advantages of using the quadratic formula are now evident.
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