If α and β are the roots of the equation x 2+ px + q = 0, then what is α 2+ β 2equal to?
p 2– 2q
Finding the Sum of Squares of Roots of a Quadratic Equation
The question asks for the value of \(\alpha^2 + \beta^2\), where \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(x^2 + px + q = 0\).
A standard quadratic equation is given by \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients, and \(x\) is the variable.
Comparing the given equation \(x^2 + px + q = 0\) with the standard form, we can identify the coefficients:
- Coefficient of \(x^2\), \(a = 1\)
- Coefficient of \(x\), \(b = p\)
- Constant term, \(c = q\)
Using Vieta's Formulas for Roots
Vieta's formulas provide relationships between the roots of a polynomial equation and its coefficients. For a quadratic equation \(ax^2 + bx + c = 0\) with roots \(\alpha\) and \(\beta\), Vieta's formulas state:
- Sum of roots: \(\alpha + \beta = -\frac{b}{a}\)
- Product of roots: \(\alpha \beta = \frac{c}{a}\)
Applying these formulas to the given equation \(x^2 + px + q = 0\):
- Sum of roots: \(\alpha + \beta = -\frac{p}{1} = -p\)
- Product of roots: \(\alpha \beta = \frac{q}{1} = q\)
So, we have \(\alpha + \beta = -p\) and \(\alpha \beta = q\).
Calculating the Sum of Squares of Roots
We need to find the value of \(\alpha^2 + \beta^2\). We can use a common algebraic identity that relates the sum of squares to the sum and product of two numbers:
\((\alpha + \beta)^2 = \alpha^2 + 2\alpha\beta + \beta^2\)
Rearranging this identity to solve for \(\alpha^2 + \beta^2\), we get:
\(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\)
Now, substitute the values of \((\alpha + \beta)\) and \((\alpha \beta)\) that we found using Vieta's formulas:
- Substitute \((\alpha + \beta) = -p\)
- Substitute \((\alpha \beta) = q\)
So, \(\alpha^2 + \beta^2 = (-p)^2 - 2(q)\).
Simplifying the expression:
\(\alpha^2 + \beta^2 = p^2 - 2q\)
Thus, the value of \(\alpha^2 + \beta^2\) is \(p^2 - 2q\).
Matching with the Options
Let's compare our result \(p^2 - 2q\) with the given options:
| Option | Expression |
|---|---|
| 1 | \(p^2 - 2q\) |
| 2 | \(q^2 - 2p\) |
| 3 | \(p^2 + 2q\) |
| 4 | \(q^2 - q\) |
Our calculated value \(\alpha^2 + \beta^2 = p^2 - 2q\) matches Option 1.
Revision Table: Quadratic Roots and Coefficients
| Concept | Formula (for \(ax^2+bx+c=0\)) |
|---|---|
| Sum of Roots (\(\alpha+\beta\)) | \(-\frac{b}{a}\) |
| Product of Roots (\(\alpha\beta\)) | \(\frac{c}{a}\) |
| Sum of Squares of Roots (\(\alpha^2+\beta^2\)) | \((\alpha+\beta)^2 - 2\alpha\beta = \left(-\frac{b}{a}\right)^2 - 2\left(\frac{c}{a}\right) = \frac{b^2}{a^2} - \frac{2c}{a} = \frac{b^2-2ac}{a^2}\) |
Additional Information: Exploring Quadratic Equation Roots
Understanding the relationship between the roots and coefficients of a quadratic equation is fundamental in algebra. This relationship is elegantly captured by Vieta's formulas.
- Discriminant: For a quadratic equation \(ax^2+bx+c=0\), the discriminant is \(\Delta = b^2 - 4ac\). The nature of the roots depends on the discriminant:
- If \(\Delta > 0\), the roots are real and distinct.
- If \(\Delta = 0\), the roots are real and equal.
- If \(\Delta < 0\), the roots are complex conjugates.
- Other Root Relationships: Besides the sum and product, other symmetric expressions of roots can be found using Vieta's formulas. For example, the sum of cubes \(\alpha^3 + \beta^3\) can be expressed as \((\alpha + \beta)(\alpha^2 - \alpha\beta + \beta^2)\) or \((\alpha+\beta)^3 - 3\alpha\beta(\alpha+\beta)\).
- Forming a Quadratic Equation: If the roots \(\alpha\) and \(\beta\) of a quadratic equation are known, the equation can be written as \(x^2 - (\alpha + \beta)x + \alpha\beta = 0\), or \(x^2 - (\text{Sum of roots})x + (\text{Product of roots}) = 0\).
These concepts are crucial for solving problems involving roots of polynomials without explicitly finding the roots themselves.
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