Analyzing Quadratic Equation Roots

The given equation is a quadratic equation: \((a + b) x^2 - (a + b + c) x + k = 0\), where \(a\), \(b\), and \(c\) are real numbers.

We are given that \(k = \frac{c}{2}\) and \(c \neq 0\).

To determine the nature of the roots of this quadratic equation, we need to examine its discriminant. A standard quadratic equation is in the form \(Ax^2 + Bx + C = 0\). Comparing this with the given equation, we can identify the coefficients:

• Coefficient of \(x^2\): \(A = a + b\)
• Coefficient of \(x\): \(B = -(a + b + c)\)
• Constant term: \(C = k = \frac{c}{2}\)

Calculating the Discriminant

The discriminant of a quadratic equation \(Ax^2 + Bx + C = 0\) is given by the formula \(\Delta = B^2 - 4AC\). Let's substitute the coefficients we found:

\[ \Delta = (-(a + b + c))^2 - 4(a + b)\left(\frac{c}{2}\right) \]

Simplify the expression:

\[ \Delta = (a + b + c)^2 - 2c(a + b) \]

Expand \((a + b + c)^2\). We know that \((x+y+z)^2 = x^2+y^2+z^2+2xy+2xz+2yz\). Let \(x = (a+b)\) and \(y=c\). So \((a+b+c)^2 = ((a+b)+c)^2 = (a+b)^2 + c^2 + 2c(a+b)\).

Substitute this back into the discriminant equation:

\[ \Delta = [(a + b)^2 + c^2 + 2c(a + b)] - 2c(a + b) \]

\[ \Delta = (a + b)^2 + c^2 + 2c(a + b) - 2c(a + b) \]

The terms \(+2c(a + b)\) and \(-2c(a + b)\) cancel out:

\[ \Delta = (a + b)^2 + c^2 \]

Determining the Nature of Roots

Now we need to analyze the value of the discriminant \(\Delta = (a + b)^2 + c^2\) to determine the nature of the roots of the quadratic equation.

We are given that \(a\), \(b\), and \(c\) are real numbers, and \(c \neq 0\).

• Since \(a\) and \(b\) are real, their sum \((a+b)\) is also real. The square of any real number is always non-negative. Thus, \((a + b)^2 \ge 0\).
• Since \(c\) is a real number and \(c \neq 0\), its square \(c^2\) is always positive. Thus, \(c^2 > 0\).

The discriminant \(\Delta\) is the sum of \((a+b)^2\) and \(c^2\).

\[ \Delta = (a + b)^2 + c^2 \]

Since \((a + b)^2 \ge 0\) and \(c^2 > 0\), their sum must be strictly positive.

\[ \Delta > 0 \]

For a quadratic equation \(Ax^2 + Bx + C = 0\), the nature of the roots is determined by the sign of the discriminant:

• If \(\Delta > 0\), the roots are real and unequal (distinct).
• If \(\Delta = 0\), the roots are real and equal (repeated).
• If \(\Delta < 0\), the roots are complex (non-real).

In our case, \(\Delta > 0\). Therefore, the roots of the given quadratic equation are real and unequal.

Summary of Root Nature

Based on the discriminant calculation, the nature of the roots for the quadratic equation \((a + b) x^2 - (a + b + c) x + k = 0\) with \(k = c/2\) (and \(c \neq 0\)) is determined by \(\Delta = (a+b)^2 + c^2\). Since \((a+b)^2 \ge 0\) and \(c^2 > 0\), \(\Delta > 0\). This implies the roots are real and unequal.

Revision Table: Quadratic Roots Analysis

Additional Information: Properties of Quadratic Equations

Understanding the properties of quadratic equations is crucial for solving problems related to their roots and graphs.

• Sum and Product of Roots: For a quadratic equation \(Ax^2 + Bx + C = 0\) with roots \(\alpha\) and \(\beta\), the sum of roots is \(\alpha + \beta = -\frac{B}{A}\) and the product of roots is \(\alpha \beta = \frac{C}{A}\).
• Relationship with Graph: The roots of a quadratic equation \(y = Ax^2 + Bx + C\) are the x-intercepts (where the graph crosses or touches the x-axis).
• Vertex: The vertex of the parabola \(y = Ax^2 + Bx + C\) is located at \(x = -\frac{B}{2A}\). The y-coordinate of the vertex gives the minimum or maximum value of the quadratic function.
• Factorization: If the roots are \(\alpha\) and \(\beta\), the quadratic expression \(Ax^2 + Bx + C\) can be factored as \(A(x - \alpha)(x - \beta)\). This is possible when the roots are real.

In this specific problem, the condition \(c \neq 0\) is essential because if \(c\) were 0, then \(k\) would be 0, and the discriminant would simply be \((a+b)^2\), which could be 0 if \(a+b=0\), leading to real and equal roots. The non-zero value of \(c\) guarantees \(c^2 > 0\) and hence \(\Delta > 0\).