Question

If \(\alpha, \beta\) are the roots of the equation \(x^{2}-p x+36=0\) and \(\alpha^{2}+\beta^{2}=9\), then the value of \(p\) is (a) \(\pm 3\) (b) \(\pm 6\) (c) \(\pm 8\) (d) \(\pm 9\)

Short answer

Answer: The value of \(p\) is \(\pm 9\).

Step-by-step solution

Express \(\alpha\) and \(\beta\) in terms of coefficients of the given equation

Using Vieta's formulas, the sum of the roots and product of the roots can be expressed in terms of coefficients of the given quadratic equation. Sum of the roots: \(\alpha + \beta = p\) Product of the roots: \(\alpha \cdot \beta = 36\)

Use Condition (\(\alpha^2 + \beta^2 = 9\))

We are given that \(\alpha^2 + \beta^2 = 9\). We can express this condition in terms of \(p\) by doing the following: Square the sum of the roots equation and subtract the product of the roots twice from it: \((\alpha + \beta)^2 - 2(\alpha \cdot \beta) = \alpha^2 + \beta^2\)

Substitute the expressions for \(\alpha + \beta\) and \(\alpha \cdot \beta\) into the equation

\(\alpha + \beta = p\) and \(\alpha \cdot \beta = 36\). Substituting these expressions into the equation from step 2, we get: \((p)^2 - 2(36) = 9\)

Solve for \(p\)

Now, solve the equation for \(p\): \(p^2 - 72 = 9\) \(p^2 = 81\) \(p = \pm \sqrt{81}\) \(p = \pm 9\) The value of \(p\) is \(\pm 9\). So, the correct answer is (d) \(\pm 9\).

Key concepts

Vieta's formulas

Vieta's formulas are a valuable tool in understanding the relationships between the roots of a polynomial equation and its coefficients. When dealing with quadratic equations, like the one in our exercise, Vieta's formulas provide a convenient way to express the sum and product of the roots without solving explicitly for the roots. For any quadratic equation of the form \( ax^2 + bx + c = 0 \), Vieta's formulas tell us:

• The sum of the roots \( \alpha + \beta = -\frac{b}{a} \).
• The product of the roots \( \alpha \cdot \beta = \frac{c}{a} \).

In our specific example where the equation is \( x^2 - px + 36 = 0 \), this changes to \( \alpha + \beta = p \) and \( \alpha \cdot \beta = 36 \). These expressions immensely simplify calculations, especially when additional conditions, like \( \alpha^2 + \beta^2 = 9 \), are provided.

Sum of roots

The sum of the roots of a quadratic equation offers insight into how the individual roots relate to each other. According to Vieta’s formulas, for a given quadratic equation of form \( x^2 - px + c = 0 \), the sum of the roots \( \alpha + \beta \) can be directly found as \( p \). This is because the sum is defined by the negative coefficient of the \( x \) term divided by the leading coefficient, typically 1 in a normalized equation, hence giving \( -(-p) = p \). Understanding how to express the sum of roots helps streamline solutions to quadratic problems where

• You need to find possible root values given conditions like \( \alpha^2 + \beta^2 = \, \text{a specific number} \).
• You simplify further calculations without expanding into more lengthy algebraic manipulation.

Product of roots

The product of the roots of a quadratic equation, as indicated by Vieta’s formulas, provides another angle to tackle problems involving quadratic roots without directly solving for them. In our equation, \( x^2 - px + 36 = 0 \), the product of the roots \( \alpha \cdot \beta \) is given by the constant term. This translates to \( 36 \) in our scenario as seen through \( c = \alpha \cdot \beta = 36 \). Being aware of the product allows:

• Easy verification of root values especially when combined with other conditions such as sum of squares.
• Checking consistencies when determining specific values or testing conditions in given problems.

The product is crucial when checking equations or forming new equations based on specific root characteristics.

Algebraic identities

Algebraic identities are formulae that hold true for all values of their variables, and they are extremely helpful when solving equations by simplifying expressions. In this task, the identity connecting squares of the roots to their sum and product was pivotal: \[ (\alpha + \beta)^2 = \alpha^2 + \beta^2 + 2\alpha\beta \] By rearranging the identity, we derived: \[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \] This algebraic identity allowed us to substitute known quantities (sum and product of roots) into the equation: \( p^2 - 72 = 9 \), which was much simpler to solve. These identities facilitate manipulation of equations by reducing the complexity and eliminating need for complex root-solving, thus saving time and effort during problem-solving.