Finding Roots of Quadratic Equation when Sum of Coefficients is Zero

We are given a quadratic equation \(ax^2 + bx + c = 0\), where \(a, b, c\) are non-zero real numbers. We are also given a special condition that the sum of the coefficients is zero, i.e., \(a + b + c = 0\).

We need to find the roots of this quadratic equation under this specific condition.

Testing for a Root: \(x=1\)

Let's consider a simple value for \(x\) and see if it satisfies the equation given the condition. Let's try \(x = 1\). Substitute \(x = 1\) into the quadratic equation:

\(a(1)^2 + b(1) + c\)

\(= a(1) + b + c\)

\(= a + b + c\)

We are given that \(a + b + c = 0\). Therefore, when \(x = 1\), the equation \(ax^2 + bx + c = 0\) becomes \(0 = 0\), which is true.

This confirms that \(x = 1\) is one of the roots of the quadratic equation \(ax^2 + bx + c = 0\) when \(a + b + c = 0\).

Using Vieta's Formulas to Find the Second Root

For a standard quadratic equation \(ax^2 + bx + c = 0\), Vieta's formulas relate the roots (\(\alpha\) and \(\beta\)) to the coefficients:

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

We have found one root, let's say \(\alpha = 1\). We can use the product of roots formula to find the second root, \(\beta\).

The product of the roots is \(1 \cdot \beta = \frac{c}{a}\).

This directly gives us the second root: \(\beta = \frac{c}{a}\).

Verifying the Roots

The roots we found are \(1\) and \(\frac{c}{a}\). Let's check if their sum is equal to \(-\frac{b}{a}\) using the given condition \(a+b+c=0\).

Sum of roots: \(1 + \frac{c}{a}\)

To compare this with \(-\frac{b}{a}\), we can rearrange the condition \(a + b + c = 0\) to express \(b\) in terms of \(a\) and \(c\):

\(b = -a - c\)

So, \(-\frac{b}{a} = -\frac{(-a - c)}{a} = \frac{a + c}{a} = \frac{a}{a} + \frac{c}{a} = 1 + \frac{c}{a}\).

The sum of the roots \(1 + \frac{c}{a}\) matches \(-\frac{b}{a}\). This further confirms that the roots are indeed \(1\) and \(\frac{c}{a}\).

Final Roots of the Equation

Given the condition \(a + b + c = 0\) for the non-zero real coefficients \(a, b, c\) of the quadratic equation \(ax^2 + bx + c = 0\), the roots are \(1\) and \(\frac{c}{a}\).

Revision Table: Quadratic Roots

Additional Information: Properties of Quadratic Equations

A quadratic equation is a polynomial equation of the second degree. Its general form is \(ax^2 + bx + c = 0\), where \(x\) is the variable, and \(a, b, c\) are coefficients, with \(a \neq 0\). The roots of a quadratic equation are the values of \(x\) that satisfy the equation.

• The roots can be found using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• The term \(b^2 - 4ac\) is called the discriminant (\(\Delta\)). It determines the nature of the roots:
  • If \(\Delta > 0\), there are two distinct real roots.
  • If \(\Delta = 0\), there is exactly one real root (a repeated root).
  • If \(\Delta < 0\), there are two distinct complex roots.
• The condition \(a+b+c=0\) provides a shortcut for finding roots without using the quadratic formula in this specific case. It essentially tells us that \(x=1\) is always a root under this condition.