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:

Our calculated value \(\alpha^2 + \beta^2 = p^2 - 2q\) matches Option 1.

Revision Table: Quadratic Roots and Coefficients

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.