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Welcome to our Math lesson on Vieta's Formulas, this is the fourth 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.
So far, we have had the quadratic equation given and the task was to find the roots (if any). Now, we are going to consider a quadratic from the inverse point of view, i.e. to find the coefficients and constant of the original equation when the two roots are known. For this, we have to use two formulas, known as Vieta's Formulas. Thus, if the roots of a quadratic equation are x1 and x2; the coefficients are a and b (as usual) and the constant of the equation is c, then Vieta's Formulas are as follows:
Proof: For the first formula, we have based on the quadratic formula
and for the second formula, we have
The two roots of a quadratic equation are - 1 and 3. What is this equation?
From Vieta's Formulas, we can write
and
Therefore, since all quadratic equations have the form
we can substitute b and c in terms of a using the relations found through Vieta's Formulas. Thus, we obtain
Dividing both sides by a yields
Hence, simplifying by a, we obtain the original quadratic equation
(You can find the proof by calculating the roots of the last equation. If you follow all steps carefully, you will obtain the root x1 = - 1 and x2 = 3, as given in the clues.)
Another example to better clarify this point:
The sum of the roots in a quadratic equation is - 7 and their product is 6. What are the roots?
From Vieta's Formulas, we have
Likewise,
Writing the quadratic equation in terms of a yields
Simplifying the coefficient a from all terms yields
Now, we can use the quadratic formula to calculate the roots. Given that a = 1, b = 7 and c = 6, we first calculate the discriminant, i.e.
Hence, the two roots are:
and
In other cases, it may be very useful to raise the clues in the second power to obtain useful info in a much shorter way than above. Vieta's Formulas are very helpful in this regard. Let's consider an example to clarify this point.
The sum of the two roots in a quadratic equation is 2 and their product is - 3. What is x12 + x22?
From the clues, we have
Raising both sides in square yields
Again, from the clues, it is known that x1 · x2 = -3. Therefore,
Hence,
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