Question

(a) The equation \(2 x^{2}+(k-1) x+2=0\) has two (p) \(\mathrm{R}-\\{-2,0,2\\}\) real roots if k lies in the interval (b) The equation \(\mathrm{kx}^{2}+9 \mathrm{x}-\mathrm{k}-2=0\) where, \(\mathrm{k}\) is real has (q) \(\left(\frac{-1}{2}, 0\right)\) complex roots if k lies in the interval (c) The equation \(x^{2}+3 x-k^{2}=0\) has irrational roots if (r) No real values for \(\mathrm{k}\) exist integer \(\mathrm{k}\) belongs to (d) The equation \(\mathrm{kx}^{2}-2 \mathrm{x}+1+2 \mathrm{k}=0\) has one (s) \((-\infty,-3] \cup[5, \infty)\) root positive and the other root negative if \(\mathrm{k}\) lies in the interval

Short answer

a) \(2x^2 + (k-1)x + 2 = 0\) has two real roots. Answer: \(k \in (-\infty, -3] \cup [5, \infty)\) b) \(kx^2 + 9x - k - 2 = 0\) has complex roots. Answer: \(k \in \left(\frac{-1}{2}, 0\right)\) c) \(x^2 + 3x - k^2 = 0\) has irrational roots. Answer: No specific interval for k with integer values found. The answer is true for all real values of k for which \(9 + 4k^2\) is not a perfect square. d) \(kx^2 - 2x + 1 + 2k = 0\) has one positive root and one negative root. Answer: \(k \in (-\infty,-\frac{1}{2})\)

Step-by-step solution

Find the discriminant of the equation

We are given the quadratic equation \(2x^2 + (k-1)x + 2 = 0\). The discriminant (\(\Delta\)) is calculated as \(\Delta = b^2 - 4ac\). In this case, \(a = 2\), \(b = k-1\), and \(c = 2\). So, \(\Delta = (k-1)^2 - 4(2)(2)\).

Determine the condition for real roots

For the equation to have two real roots, we need \(\Delta \geq 0\). Thus, \((k-1)^2 - 4(2)(2) \geq 0\). Simplifying, we get \((k-1)^2 \geq 16\).

Find the interval for k

To solve for k, we take the square root of both sides: \(|k-1| \geq 4\). This gives us two possible intervals for k: \(k-1 \geq 4\) or \(k-1 \leq -4\). Solving both inequalities, we get the intervals \(k \in (-\infty, -3] \cup [5, \infty)\). So, the answer for (a) is \((p) \in (-\infty, -3] \cup [5, \infty)\). #b)#

Find the discriminant of the equation

We are given the quadratic equation \(kx^2 + 9x - k - 2 = 0\). The discriminant (\(\Delta\)) is calculated as \(\Delta = b^2 - 4ac\). In this case, \(a = k\), \(b = 9\), and \(c = -k-2\). So, \(\Delta = 9^2 - 4(k)(-k-2)\).

Determine the condition for complex roots

For the equation to have complex roots, we need \(\Delta < 0\). Thus, \(81 - 4k^2 + 8k < 0\).

Find the interval for k

Solving the inequality \(81 - 4k^2 + 8k < 0\), we get the interval for k as \(k \in \left(\frac{-1}{2}, 0\right)\). So, the answer for (b) is \((q) \in \left(\frac{-1}{2}, 0\right)\). #c)#

Find the discriminant of the equation

We are given the quadratic equation \(x^2 + 3x - k^2 = 0\). The discriminant (\(\Delta\)) is calculated as \(\Delta = b^2 - 4ac\). In this case, \(a = 1\), \(b = 3\), and \(c = -k^2\). So, \(\Delta = 3^2 - 4(1)(-k^2)\).

Determine the condition for irrational roots

For the equation to have irrational roots, we need \(\Delta\) to be a non-perfect square greater than 0: \(\Delta > 0\) and it's not a perfect square. Thus, \(9 + 4k^2 > 0\) and \(9 + 4k^2\) is not a perfect square.

Find the values for k

Since \(9 + 4k^2 > 0\) for all real values of k, we just need to find the values for k where \(9 + 4k^2\) is not a perfect square. However, it is not possible to specify a precise interval for k in this case. So, we cannot find a specific set of possible values of integer k (r) for the given condition. Instead, this answer should be interpreted as being true for all real values of k for which \(9 + 4k^2\) is not a perfect square. #d)#

Determine the condition for one positive root and one negative root

If the quadratic \(kx^2 - 2x + 1 + 2k = 0\) has one positive root and one negative root, the product of the roots (\(\frac{c}{a}\)) should be negative. In this case, \(a = k\), \(c = 1+2k\). So, we need \(\frac{1+2k}{k} < 0\).

Find the interval for k

To solve for the inequality \(\frac{1+2k}{k} < 0\), we can consider the following cases: 1. \(k > 0\), then \(1+2k > 0\) and \(\frac{1+2k}{k} > 0\), which does not satisfy the inequality. 2. \(k < 0\), then \(1+2k < 0\) and \(\frac{1+2k}{k} > 0\), which satisfies the inequality. So, the answer for (d) is \((s) \in (-\infty,-\frac{1}{2})\). For each of the parts, we have now found the interval of k satisfying the given conditions: a) \(k \in (-\infty, -3] \cup [5, \infty)\) b) \(k \in \left(\frac{-1}{2}, 0\right)\) c) No specific interval for k with integer values found. d) \(k \in (-\infty,-\frac{1}{2})\)

Key concepts

Discriminant

The discriminant is a key component in understanding how many and what type of roots a quadratic equation will have. It's part of the quadratic formula and is represented as \( \Delta = b^2 - 4ac \). This formula comes from the general quadratic equation \( ax^2 + bx + c = 0 \). By calculating \( \Delta \), you determine the nature of the roots without actually solving for them.

The value of the discriminant tells us the nature of the roots:

• If \( \Delta > 0 \), the quadratic equation has two distinct real roots.
• If \( \Delta = 0 \), the equation has exactly one real root, which is also a repeated root.
• If \( \Delta < 0 \), the roots are complex, meaning they have imaginary components.

Understanding the discriminant helps in quickly determining the type of solutions an equation has, especially in standardized tests or in systematic analysis.

Real Roots

Real roots occur when a quadratic equation's discriminant is greater than or equal to zero (\( \Delta \geq 0 \)). When \( \Delta > 0 \), it indicates two distinct real roots.

On the other hand, when \( \Delta = 0 \), the equation has a single real root which is also a repeated or double root.

This means that the graph of the quadratic function touches the x-axis at exactly one point. Real roots are crucial as they represent the x-values where the graph intersects or touches the x-axis, showing practical solutions to the quadratic equation.

Complex Roots

Complex roots are encountered when the discriminant of a quadratic equation is less than zero (\( \Delta < 0 \)). This indicates that the quadratic equation does not intersect the x-axis, leading to roots that include imaginary components.

Complex roots always occur in conjugate pairs. For example, if one of the roots is \( a + bi \), the conjugate pair will be \( a - bi \). Complex roots arise in scenarios where the solutions to the quadratic equation are not purely real, often seen in advanced mathematics and physics problems.
These roots cannot be plotted on a traditional two-dimensional graph but are vital for understanding behaviors in complex planes or systems.

Irrational Roots

Irrational roots occur when a quadratic equation's solutions are real numbers that cannot be expressed as a simple fraction. This happens when the discriminant is a positive number, but not a perfect square. Because it's not a perfect square, the result of the quadratic formula involves square roots that are irrational.

An equation like \( x^2 - 2 = 0 \) has irrational roots, as solving it yields \( \pm\sqrt{2} \), which are numbers that cannot be plainly written as fractions. Identifying irrational roots helps in understanding the precision of the roots' representation, as they usually require maintaining significant digits or approximations.

Root Intervals

Root intervals are important for determining which specific values or ranges of values for certain parameters satisfy a given condition for quadratic equations. When solving quadratic equations, especially when a parameter like \( k \) affects the equation, root intervals help identify the valid range for \( k \).

For example, to have real roots, the inequality derived from the discriminant can provide intervals for which these conditions hold true. Similarly, obtaining complex or irrational roots requires solving through inequalities to find intervals of acceptable values for the parameters.

Using root intervals, mathematicians can determine the specific intervals where a quadratic holds certain properties, allowing for targeted analysis or predictions about family behaviors or real-world phenomenon.