Introduction to Quadratic Equations
A quadratic equation is a polynomial equation of the second degree, generally expressed as:
x2 bx c 0
where b and c are coefficients, and x is the variable. The roots of this equation, denoted as α (alpha) and β (beta), can be found using various mathematical techniques, including Vieta's formulas, which provide relationships between the coefficients of the polynomial and its roots.
Vieta's Formulas
Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. For the quadratic equation:
x2 bx c 0
the formulas are:
The sum of the roots: α β -b The product of the roots: αβ cSolving Quadratic Equations with Given Conditions
Consider the quadratic equation:
x2 5x a 0
We have the following conditions:
The sum of the roots: α β -5 The product of the roots: αβ a 2α 5β -1 (an additional condition)To solve for α (alpha) and β (beta) and find the value of a, we will use these given conditions step by step.
Expressing β in Terms of α
From the sum of the roots, we can express β in terms of α:
α β -5
Subtracting α from both sides, we get:
β -5 - α
Solving for α and β
We substitute β -5 - α into the additional condition:
2α 5(-5 - α) -1
Expanding the equation:
2α - 25 - 5α -1
Combining like terms:
-3α - 25 -1
Adding 25 to both sides:
-3α 24
Dividing both sides by -3:
α -8
Finding β
Now, we substitute α -8 back into the equation β -5 - α:
β -5 - (-8) -5 8 3
Calculating a
The product of the roots αβ a:
αβ (-8) × 3 -24
Therefore, the value of a is:
boxed{-24}
Conclusion
In this article, we have demonstrated how to solve quadratic equations using the given conditions and Vieta's formulas. The value of the variable 'a' was derived by first expressing one root in terms of the other and then substituting back into the given conditions. This approach is a fundamental technique in algebra that helps in solving complex quadratic problems.