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The qudratic equation x2 + 15 |x| + 14 = 0 has
- a)only positive solutions
- b)only negative solutions
- c)no solution
- d)both positive and negative solution
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
The qudratic equation x2 + 15 |x| + 14 = 0 hasa)only positive solution...
Solution:
Given: x2 + 15 |x| + 14 = 0
We can factorize the given quadratic equation as:
(x + 14)(x + 1) + 15 |x| = 0
Now, we can split the equation into two cases:
Case 1: x ≥ 0
In this case, |x| = x, so we have:
(x + 14)(x + 1) + 15x = 0
Expanding the expression and simplifying, we get:
x2 + 30x + 14 = 0
Using the quadratic formula, we get:
x = (-30 ± √(302 - 4(14)))/2
x = (-30 ± √728)/2
x = -15 ± √182
Since x ≥ 0, we can reject the negative root and take:
x = -15 + √182
Case 2: x < />
In this case, |x| = -x, so we have:
(x + 14)(x + 1) - 15x = 0
Expanding the expression and simplifying, we get:
x2 + 16x + 14 = 0
Using the quadratic formula, we get:
x = (-16 ± √(162 - 4(14)))/2
x = (-16 ± √132)/2
x = -8 ± √33
Since x < 0,="" we="" can="" reject="" the="" positive="" root="" and="" />
x = -8 - √33
Therefore, the given quadratic equation has no real solutions as both the roots are of opposite signs. Hence, the correct answer is option C.
Given: x2 + 15 |x| + 14 = 0
We can factorize the given quadratic equation as:
(x + 14)(x + 1) + 15 |x| = 0
Now, we can split the equation into two cases:
Case 1: x ≥ 0
In this case, |x| = x, so we have:
(x + 14)(x + 1) + 15x = 0
Expanding the expression and simplifying, we get:
x2 + 30x + 14 = 0
Using the quadratic formula, we get:
x = (-30 ± √(302 - 4(14)))/2
x = (-30 ± √728)/2
x = -15 ± √182
Since x ≥ 0, we can reject the negative root and take:
x = -15 + √182
Case 2: x < />
In this case, |x| = -x, so we have:
(x + 14)(x + 1) - 15x = 0
Expanding the expression and simplifying, we get:
x2 + 16x + 14 = 0
Using the quadratic formula, we get:
x = (-16 ± √(162 - 4(14)))/2
x = (-16 ± √132)/2
x = -8 ± √33
Since x < 0,="" we="" can="" reject="" the="" positive="" root="" and="" />
x = -8 - √33
Therefore, the given quadratic equation has no real solutions as both the roots are of opposite signs. Hence, the correct answer is option C.
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