If the roots of the equation x2 - bx + c = 5 differ by 5, then which one of the following is correct ?
b2 = 4c + 5
Analyzing the Quadratic Equation and Root Properties
The problem asks us to find a relationship between the coefficients b and c of a quadratic equation, given a specific condition about its roots. The equation provided is \(x^2 - bx + c = 5\). A key piece of information is that the roots of this equation differ by 5.
Standard Form of the Quadratic Equation
First, let's rewrite the given equation in the standard quadratic form, which is \(ax^2 + Bx + C = 0\). Subtracting 5 from both sides gives us:
\(x^2 - bx + (c - 5) = 0\)
In this standard form, we have \(a = 1\), \(B = -b\), and \(C = (c - 5)\).
Applying Vieta's Formulas
Vieta's formulas relate the coefficients of a polynomial to the sums and products of its roots. For a quadratic equation \(ax^2 + Bx + C = 0\), let the roots be \(\alpha\) and \(\beta\). The formulas are:
- Sum of roots: \(\alpha + \beta = -B/a\)
- Product of roots: \(\alpha \beta = C/a\)
Applying these to our equation \(x^2 - bx + (c - 5) = 0\):
- Sum of roots: \(\alpha + \beta = -(-b)/1 = b\)
- Product of roots: \(\alpha \beta = (c - 5)/1 = c - 5\)
Using the Condition on the Difference of Roots
The question states that the roots differ by 5. We can express this condition mathematically as:
\(|\alpha - \beta| = 5\)
Squaring both sides, we get:
\((\alpha - \beta)^2 = 5^2 = 25\)
Relating Sum, Product, and Difference of Roots
There is a useful identity that connects the square of the difference of roots to the sum and product of the roots:
\((\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta\)
Now, we can substitute the expressions for the sum and product of roots that we found using Vieta's formulas:
- \(\alpha + \beta = b\)
- \(\alpha \beta = c - 5\)
- \((\alpha - \beta)^2 = 25\)
Substituting these into the identity:
\(25 = (b)^2 - 4(c - 5)\)
Deriving the Final Relationship
Let's simplify the equation:
\(25 = b^2 - 4(c - 5)\)
Distribute the -4:
\(25 = b^2 - 4c + 20\)
Now, we want to isolate \(b^2\) to match the format of the options. Subtract 20 from both sides:
\(25 - 20 = b^2 - 4c\)
\(5 = b^2 - 4c\)
Finally, add \(4c\) to both sides to get the desired form:
\(b^2 = 4c + 5\)
Conclusion
The relationship derived, \(b^2 = 4c + 5\), matches one of the provided options. This equation holds true if the roots of the quadratic equation \(x^2 - bx + c = 5\) differ by 5.
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