Understanding the Quadratic Equation and its Roots

A standard quadratic equation is given in the form \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are coefficients and \( a \neq 0 \). The solutions to this equation are called its roots. Let's denote the roots of the equation \( ax^2 + bx + c = 0 \) as \( \alpha \) and \( \beta \).

Sum and Product of Roots Formulas

For a quadratic equation \( ax^2 + bx + c = 0 \), there are well-known formulas relating the roots \( \alpha \) and \( \beta \) to the coefficients \( a, b, \) and \( c \):

• Sum of the roots: \( \alpha + \beta = -\frac{b}{a} \)
• Product of the roots: \( \alpha \beta = \frac{c}{a} \)

Applying the Given Condition for Quadratic Roots

The question states that the sum of the roots is equal to the sum of their squares. We can write this condition mathematically using the roots \( \alpha \) and \( \beta \):

\( \alpha + \beta = \alpha^2 + \beta^2 \)

Deriving the Relationship between Coefficients \( a, b, c \)

We need to express the sum of squares of the roots \( \alpha^2 + \beta^2 \) in terms of the sum \( \alpha + \beta \) and the product \( \alpha \beta \). A useful algebraic identity is:

\( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta \)

Now, substitute the formulas for the sum and product of roots into this identity:

\( \alpha^2 + \beta^2 = \left(-\frac{b}{a}\right)^2 - 2\left(\frac{c}{a}\right) \)

\( \alpha^2 + \beta^2 = \frac{b^2}{a^2} - \frac{2c}{a} \)

The given condition is \( \alpha + \beta = \alpha^2 + \beta^2 \). Substitute the expressions for both sides:

\( -\frac{b}{a} = \frac{b^2}{a^2} - \frac{2c}{a} \)

To eliminate the denominators, we can multiply the entire equation by \( a^2 \) (since \( a \neq 0 \)).

\( a^2 \left(-\frac{b}{a}\right) = a^2 \left(\frac{b^2}{a^2}\right) - a^2 \left(\frac{2c}{a}\right) \)

\( -ab = b^2 - 2ac \)

We want to find a relationship between \( a, b, \) and \( c \). Let's rearrange the terms to match the format of the options:

Add \( 2ac \) to both sides:

\( -ab + 2ac = b^2 \)

Add \( ab \) to both sides:

\( 2ac = b^2 + ab \)

This can also be written as:

\( ab + b^2 = 2ac \)

Matching the Derived Condition with Options

Let's compare the relationship we derived, \( ab + b^2 = 2ac \), with the given options:

1. \( a^2+ b^2= c^2 \)
2. \( a^2+ b^2= a + b \)
3. \( ab + b^2= 2ac \)
4. \( ab - b^2= 2ac \)

The derived relationship \( ab + b^2 = 2ac \) exactly matches the third option.

Revision Table: Quadratic Equation Roots

Additional Information: Properties of Quadratic Equations

Beyond the sum and product of roots, quadratic equations have other important properties:

• Discriminant: The discriminant is given by \( \Delta = b^2 - 4ac \). It tells us about the nature of the roots:
  • If \( \Delta > 0 \), the roots are real and distinct.
  • If \( \Delta = 0 \), the roots are real and equal (a repeated root).
  • If \( \Delta < 0 \), the roots are complex conjugates.
• Quadratic Formula: The roots \( \alpha, \beta \) can be found directly using the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Forming a Quadratic Equation: If the roots \( \alpha \) and \( \beta \) are known, the quadratic equation can be written as \( x^2 - (\alpha + \beta)x + \alpha \beta = 0 \). Multiplying by a constant \( k \neq 0 \) gives \( k(x^2 - (\alpha + \beta)x + \alpha \beta) = 0 \), which can be expanded to match the \( ax^2 + bx + c = 0 \) form where \( a=k, b=-k(\alpha+\beta), c=k(\alpha\beta) \).