Which one of the following equations does not have real roots ?
4x 2 + 9x + 6 = 0
Understanding Real Roots of Quadratic Equations
A quadratic equation has the general form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a \neq 0 \). The nature of the roots (solutions) of a quadratic equation depends on the value of the discriminant, which is denoted by the Greek letter delta (\( \Delta \)) and calculated using the formula:
\( \Delta = b^2 - 4ac \)
Based on the value of the discriminant, we can determine if the quadratic equation has real roots:
- If \( \Delta > 0 \), the equation has two distinct real roots.
- If \( \Delta = 0 \), the equation has exactly one real root (or two equal real roots).
- If \( \Delta < 0 \), the equation has no real roots (it has two complex conjugate roots).
To find which of the given quadratic equations does not have real roots, we need to calculate the discriminant \( \Delta \) for each equation and check if \( \Delta < 0 \).
Analyzing Each Quadratic Equation Option
Option 1: \( 2x^2 + 16x + 3 = 0 \)
In this equation, the coefficients are \( a = 2 \), \( b = 16 \), and \( c = 3 \).
Let's calculate the discriminant:
\( \Delta = b^2 - 4ac \)
\( \Delta = (16)^2 - 4(2)(3) \)
\( \Delta = 256 - 24 \)
\( \Delta = 232 \)
Since \( \Delta = 232 > 0 \), this equation has two distinct real roots.
Option 2: \( 2x^2 + 10x - 1 = 0 \)
In this equation, the coefficients are \( a = 2 \), \( b = 10 \), and \( c = -1 \).
Let's calculate the discriminant:
\( \Delta = b^2 - 4ac \)
\( \Delta = (10)^2 - 4(2)(-1) \)
\( \Delta = 100 + 8 \)
\( \Delta = 108 \)
Since \( \Delta = 108 > 0 \), this equation has two distinct real roots.
Option 3: \( x^2 - 8x + 1 = 0 \)
In this equation, the coefficients are \( a = 1 \), \( b = -8 \), and \( c = 1 \).
Let's calculate the discriminant:
\( \Delta = b^2 - 4ac \)
\( \Delta = (-8)^2 - 4(1)(1) \)
\( \Delta = 64 - 4 \)
\( \Delta = 60 \)
Since \( \Delta = 60 > 0 \), this equation has two distinct real roots.
Option 4: \( 4x^2 + 9x + 6 = 0 \)
In this equation, the coefficients are \( a = 4 \), \( b = 9 \), and \( c = 6 \).
Let's calculate the discriminant:
\( \Delta = b^2 - 4ac \)
\( \Delta = (9)^2 - 4(4)(6) \)
\( \Delta = 81 - 96 \)
\( \Delta = -15 \)
Since \( \Delta = -15 < 0 \), this equation has no real roots.
Conclusion
Based on the discriminant values calculated for each quadratic equation, the equation \( 4x^2 + 9x + 6 = 0 \) is the only one with a negative discriminant (\( \Delta = -15 \)). Therefore, this equation does not have real roots; it has complex roots.
| Equation | a | b | c | Discriminant \( \Delta = b^2 - 4ac \) | Nature of Roots |
|---|---|---|---|---|---|
| \( 2x^2 + 16x + 3 = 0 \) | 2 | 16 | 3 | \( 16^2 - 4(2)(3) = 256 - 24 = 232 \) | Real and Distinct (\( \Delta > 0 \)) |
| \( 2x^2 + 10x - 1 = 0 \) | 2 | 10 | -1 | \( 10^2 - 4(2)(-1) = 100 + 8 = 108 \) | Real and Distinct (\( \Delta > 0 \)) |
| \( x^2 - 8x + 1 = 0 \) | 1 | -8 | 1 | \( (-8)^2 - 4(1)(1) = 64 - 4 = 60 \) | Real and Distinct (\( \Delta > 0 \)) |
| \( 4x^2 + 9x + 6 = 0 \) | 4 | 9 | 6 | \( 9^2 - 4(4)(6) = 81 - 96 = -15 \) | No Real Roots (\( \Delta < 0 \)) |
Revision Table: Discriminant and Real Roots
This table summarizes how the discriminant determines the nature of roots for a quadratic equation \( ax^2 + bx + c = 0 \).
| Discriminant \( \Delta \) | Nature of Roots |
|---|---|
| \( \Delta > 0 \) | Two distinct real roots |
| \( \Delta = 0 \) | One real root (repeated) |
| \( \Delta < 0 \) | No real roots (Two complex conjugate roots) |
Additional Information on Quadratic Roots
When a quadratic equation has no real roots (i.e., \( \Delta < 0 \)), its roots are complex numbers. These complex roots always appear as a conjugate pair. The quadratic formula provides the roots:
\( x = \frac{-b \pm \sqrt{\Delta}}{2a} \)
If \( \Delta < 0 \), we can write \( \sqrt{\Delta} = \sqrt{-1 \cdot |\Delta|} = \sqrt{-1} \cdot \sqrt{|\Delta|} = i \sqrt{|\Delta|} \), where \( i \) is the imaginary unit (\( i^2 = -1 \)). The roots then become:
\( x = \frac{-b \pm i \sqrt{|\Delta|}}{2a} \)
These are the complex conjugate roots, having the form \( p \pm qi \), where \( p = \frac{-b}{2a} \) and \( q = \frac{\sqrt{|\Delta|}}{2a} \).
Similar Questions
For what values of k, the roots of 9x 2 + 8kx + 16 = 0 are real and equal?
If the sum of the squares of the roots of the equation x 2- 14x + k = 0 is 100, then what is the value of k ?
Consider a question and two statements:
Question :
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Important Questions from Quadratic Equation
For what values of k, the roots of 9x 2 + 8kx + 16 = 0 are real and equal?
If the sum of the squares of the roots of the equation x 2- 14x + k = 0 is 100, then what is the value of k ?
- If \(\rm \left( \frac{x}{x+1} \right)^2 -5 \left( \frac{x}{x+1} \right) +6=0 \) , then the value of \(\rm \left( 1+\frac{1}{x} \right) \) is equal to :
The nature of the roots of the equation 4x 2 - 2x - 3 = 0.
If the roots of the equation (q – r)x 2+ (r – p)x + (p – q) = 0 are equal, then which of the following is true?
