If the sum of the squares of the roots of the equation x 2- 14x + k = 0 is 100, then what is the value of k ?
48
Solving for k in a Quadratic Equation given Sum of Squares of Roots
The problem asks us to find the value of the constant 'k' in the quadratic equation \(x^2 - 14x + k = 0\), given that the sum of the squares of its roots is 100.
Let the roots of the quadratic equation \(x^2 - 14x + k = 0\) be \(\alpha\) and \(\beta\).
A standard quadratic equation is given by \(ax^2 + bx + c = 0\).
Comparing the given equation \(x^2 - 14x + k = 0\) with the standard form, we can identify the coefficients:
- \(a = 1\)
- \(b = -14\)
- \(c = k\)
For a quadratic equation \(ax^2 + bx + c = 0\), the sum of the roots (\(\alpha + \beta\)) and the product of the roots (\(\alpha\beta\)) are related to the coefficients by the following formulas:
- Sum of roots: \(\alpha + \beta = -\frac{b}{a}\)
- Product of roots: \(\alpha\beta = \frac{c}{a}\)
Using these formulas for the given equation \(x^2 - 14x + k = 0\):
- Sum of roots: \(\alpha + \beta = -\frac{-14}{1} = 14\)
- Product of roots: \(\alpha\beta = \frac{k}{1} = k\)
We are given that the sum of the squares of the roots is 100. This means:
\(\alpha^2 + \beta^2 = 100\)
We know a useful algebraic identity relating the sum of squares to the sum and product of two numbers:
\(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\)
Now, we can substitute the values we found for the sum of roots and the product of roots into this identity:
\(100 = (14)^2 - 2(k)\)
Let's simplify and solve for \(k\):
\(100 = 196 - 2k\)
To isolate the term with \(k\), subtract 196 from both sides of the equation:
\(100 - 196 = -2k\)
\(-96 = -2k\)
Now, divide both sides by -2 to find the value of \(k\):
\(k = \frac{-96}{-2}\)
\(k = 48\)
Thus, the value of \(k\) is 48.
| Concept | Formula | Value for \(x^2 - 14x + k = 0\) |
|---|---|---|
| Sum of roots (\(\alpha + \beta\)) | \(-\frac{b}{a}\) | \(14\) |
| Product of roots (\(\alpha\beta\)) | \(\frac{c}{a}\) | \(k\) |
| Sum of squares (\(\alpha^2 + \beta^2\)) | \((\alpha + \beta)^2 - 2\alpha\beta\) | \(100\) |
Using the relationship \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\):
\(100 = (14)^2 - 2k\)
\(100 = 196 - 2k\)
\(2k = 196 - 100\)
\(2k = 96\)
\(k = 48\)
Revision Table: Quadratic Equation Roots and Coefficients
| Equation: \(ax^2 + bx + c = 0\) | Relationship |
|---|---|
| Sum of roots (\(\alpha + \beta\)) | \(-\frac{b}{a}\) |
| Product of roots (\(\alpha\beta\)) | \(\frac{c}{a}\) |
| Sum of squares of roots (\(\alpha^2 + \beta^2\)) | \((\alpha + \beta)^2 - 2\alpha\beta\) |
Additional Information: Solving Quadratic Equations
A quadratic equation \(ax^2 + bx + c = 0\) can be solved to find its roots using the quadratic formula:
\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
The term inside the square root, \(b^2 - 4ac\), is called the discriminant (\(\Delta\)). It determines the nature of the roots:
- If \(\Delta > 0\), there are two distinct real roots.
- If \(\Delta = 0\), there is exactly one real root (a repeated root).
- If \(\Delta < 0\), there are two complex conjugate roots.
In this problem, we used the relationships between roots and coefficients, which is often a quicker method when dealing with sums or products of roots without needing to find the roots explicitly.
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