Get comprehensive solutions to your questions with the help of IDNLearn.com's experts. Get the information you need from our community of experts who provide accurate and thorough answers to all your questions.
Sagot :
To determine the range of possible values of [tex]\( k \)[/tex] such that the equation [tex]\( k x^2 + 6k x + 5 = 0 \)[/tex] has no real roots, let's follow these steps:
1. Understand the condition for no real roots:
For a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] to have no real roots, its discriminant must be less than zero. The discriminant [tex]\( \Delta \)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
2. Identify coefficients in the given quadratic equation:
The given equation is [tex]\( k x^2 + 6k x + 5 = 0 \)[/tex]. Here, the coefficients are:
[tex]\[ a = k, \quad b = 6k, \quad c = 5 \][/tex]
3. Compute the discriminant:
Substitute [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the discriminant formula:
[tex]\[ \Delta = (6k)^2 - 4(k)(5) \][/tex]
[tex]\[ \Delta = 36k^2 - 20k \][/tex]
4. Set up the inequality for no real roots:
For the quadratic equation to have no real roots, the discriminant must be less than zero:
[tex]\[ 36k^2 - 20k < 0 \][/tex]
5. Solve the inequality:
Factorize the quadratic expression:
[tex]\[ 36k^2 - 20k = 4k(9k - 5) \][/tex]
Hence, the inequality becomes:
[tex]\[ 4k(9k - 5) < 0 \][/tex]
Since [tex]\( k \)[/tex] is a non-zero constant and must be positive:
[tex]\[ 4k(9k - 5) < 0 \][/tex]
Analyze the factor [tex]\( 4k \)[/tex]:
- [tex]\( 4k \)[/tex] is always positive if [tex]\( k > 0 \)[/tex].
Analyze the factor [tex]\( (9k - 5) \)[/tex]:
- [tex]\( (9k - 5) < 0 \)[/tex] when [tex]\( 9k < 5 \)[/tex].
Therefore, the inequality [tex]\( 4k(9k - 5) < 0 \)[/tex] holds true when:
[tex]\[ 9k - 5 < 0 \implies k < \frac{5}{9} \][/tex]
6. Conclusion:
Combine the conditions found above. Since [tex]\( k \)[/tex] is a non-zero, positive constant:
[tex]\[ 0 < k < \frac{5}{9} \][/tex]
Thus, the range of possible values of [tex]\( k \)[/tex] such that the equation [tex]\( k x^2 + 6k x + 5 = 0 \)[/tex] has no real roots is:
[tex]\[ 0 < k < \frac{5}{9} \][/tex]
1. Understand the condition for no real roots:
For a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] to have no real roots, its discriminant must be less than zero. The discriminant [tex]\( \Delta \)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
2. Identify coefficients in the given quadratic equation:
The given equation is [tex]\( k x^2 + 6k x + 5 = 0 \)[/tex]. Here, the coefficients are:
[tex]\[ a = k, \quad b = 6k, \quad c = 5 \][/tex]
3. Compute the discriminant:
Substitute [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the discriminant formula:
[tex]\[ \Delta = (6k)^2 - 4(k)(5) \][/tex]
[tex]\[ \Delta = 36k^2 - 20k \][/tex]
4. Set up the inequality for no real roots:
For the quadratic equation to have no real roots, the discriminant must be less than zero:
[tex]\[ 36k^2 - 20k < 0 \][/tex]
5. Solve the inequality:
Factorize the quadratic expression:
[tex]\[ 36k^2 - 20k = 4k(9k - 5) \][/tex]
Hence, the inequality becomes:
[tex]\[ 4k(9k - 5) < 0 \][/tex]
Since [tex]\( k \)[/tex] is a non-zero constant and must be positive:
[tex]\[ 4k(9k - 5) < 0 \][/tex]
Analyze the factor [tex]\( 4k \)[/tex]:
- [tex]\( 4k \)[/tex] is always positive if [tex]\( k > 0 \)[/tex].
Analyze the factor [tex]\( (9k - 5) \)[/tex]:
- [tex]\( (9k - 5) < 0 \)[/tex] when [tex]\( 9k < 5 \)[/tex].
Therefore, the inequality [tex]\( 4k(9k - 5) < 0 \)[/tex] holds true when:
[tex]\[ 9k - 5 < 0 \implies k < \frac{5}{9} \][/tex]
6. Conclusion:
Combine the conditions found above. Since [tex]\( k \)[/tex] is a non-zero, positive constant:
[tex]\[ 0 < k < \frac{5}{9} \][/tex]
Thus, the range of possible values of [tex]\( k \)[/tex] such that the equation [tex]\( k x^2 + 6k x + 5 = 0 \)[/tex] has no real roots is:
[tex]\[ 0 < k < \frac{5}{9} \][/tex]
We are delighted to have you as part of our community. Keep asking, answering, and sharing your insights. Together, we can create a valuable knowledge resource. IDNLearn.com provides the answers you need. Thank you for visiting, and see you next time for more valuable insights.
