Understanding the Quadratic Equation and Its Roots

The given equation is \(x(x + 1) + 1 = 0\). To find the roots and their properties, we first need to rewrite this equation in the standard quadratic form, which is \(ax^2 + bx + c = 0\).

Expanding the given equation:

\(\qquad x \cdot x + x \cdot 1 + 1 = 0\)

\(\qquad x^2 + x + 1 = 0\)

Now, the equation is in the standard form \(ax^2 + bx + c = 0\). By comparing \(x^2 + x + 1 = 0\) with the standard form, we can identify the coefficients:

• \(a = 1\) (coefficient of \(x^2\))
• \(b = 1\) (coefficient of \(x\))
• \(c = 1\) (constant term)

Properties of Roots of a Quadratic Equation

For a quadratic equation \(ax^2 + bx + c = 0\), there are well-known relationships between the roots (let's call them \(r_1\) and \(r_2\)) and the coefficients \(a, b, c\).

• Sum of roots: \(r_1 + r_2 = -\frac{b}{a}\)
• Product of roots: \(r_1 \cdot r_2 = \frac{c}{a}\)

Finding the Other Root Using Root Properties

We are given that \(k\) is one of the roots of the equation \(x^2 + x + 1 = 0\). Let the other root be \(r_2\). Using the properties mentioned above with \(a=1, b=1, c=1\):

• Sum of roots: \(k + r_2 = -\frac{1}{1} = -1\)
• Product of roots: \(k \cdot r_2 = \frac{1}{1} = 1\)

From the product of roots relationship, we have \(k \cdot r_2 = 1\). If \(k\) is a root, it must satisfy the original equation, \(k^2 + k + 1 = 0\). The roots of this specific equation \(x^2+x+1=0\) are complex and non-zero, so \(k \neq 0\). We can divide by \(k\) to find \(r_2\) in terms of \(k\):

\(\qquad r_2 = \frac{1}{k}\)

Now, let's look at the equation \(k^2 + k + 1 = 0\), which \(k\) satisfies. We can rearrange this equation to find relationships involving \(k^2\) or \(1/k\).

From \(k^2 + k + 1 = 0\), we can isolate \(k^2\):

\(\qquad k^2 = -k - 1\)

Also, from \(k^2 + k + 1 = 0\), since \(k \neq 0\), we can divide the entire equation by \(k\):

\(\qquad \frac{k^2}{k} + \frac{k}{k} + \frac{1}{k} = 0\)

\(\qquad k + 1 + \frac{1}{k} = 0\)

Now, isolate \(1/k\):

\(\qquad \frac{1}{k} = -k - 1\)

Comparing the expressions for \(k^2\) and \(1/k\), we see that both are equal to \(-k-1\). Therefore:

\(\qquad k^2 = \frac{1}{k}\)

Since we established that the other root \(r_2\) is equal to \(1/k\), we can conclude that the other root \(r_2\) is equal to \(k^2\).

Let's quickly check this using the sum of roots: \(k + r_2 = -1\). If \(r_2 = k^2\), then \(k + k^2 = -1\). Rearranging this gives \(k^2 + k + 1 = 0\), which is true because \(k\) is a root of this equation. This confirms our finding.

Analyzing the Options

Based on our derivation, the other root must be equal to \(k^2\). Let's compare this with the given options:

1. 1
2. -k
3. k²
4. -k²

Option 3, \(k^2\), matches our derived value for the other root.

Revision Table: Quadratic Equation Roots

Additional Information: The Roots of \(x^2 + x + 1 = 0\)

The roots of the equation \(x^2 + x + 1 = 0\) are special. Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):

\(\qquad x = \frac{-1 \pm \sqrt{1^2 - 4(1)(1)}}{2(1)} = \frac{-1 \pm \sqrt{1 - 4}}{2} = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 \pm i\sqrt{3}}{2}\)

These roots are complex numbers, often denoted by \(\omega\) and \(\omega^2\). They are the non-real cube roots of unity.

• One root is \(\omega = \frac{-1 + i\sqrt{3}}{2}\).
• The other root is \(\omega^2 = \left(\frac{-1 + i\sqrt{3}}{2}\right)^2 = \frac{1 - 3 - 2i\sqrt{3}}{4} = \frac{-2 - 2i\sqrt{3}}{4} = \frac{-1 - i\sqrt{3}}{2}\).

Notice that if one root is \(k = \omega\), the other root is \(\omega^2 = k^2\). If one root is \(k = \omega^2\), the other root is \(\omega = (\omega^2)^2 = \omega^4 = \omega^3 \cdot \omega = 1 \cdot \omega = \omega\). But wait, if \(k=\omega^2\), is the other root \(k^2\)? \(k^2 = (\omega^2)^2 = \omega^4 = \omega\). So yes, if \(k=\omega^2\), the other root \(\omega\) is indeed \(k^2\). This confirms that if \(k\) is one root, the other root is \(k^2\). These roots also satisfy \(\omega^3 = 1\). If \(k\) is a root, \(k^3=1\). The product of roots is \(k \cdot r_2 = 1\). Since \(k^3 = 1\), we have \(k \cdot k^2 = 1\). Comparing \(k \cdot r_2 = 1\) and \(k \cdot k^2 = 1\), we see \(r_2 = k^2\). This provides another way to confirm the result.