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For what value of 'm' will the quadratic equation x2 - mx + 4 = 0 have real and equal roots?
- a)16
- b)8
- c)2
- d)-4
- e)Choice (B) and (C)
Correct answer is option 'D'. Can you explain this answer?
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For what value of m will the quadratic equation x2 - mx + 4 = 0 have r...
Real and Equal Roots of a Quadratic Equation
To find the value of \( m \) for which the quadratic equation \( x^2 - mx + 4 = 0 \) has real and equal roots, we need to determine when the discriminant of the quadratic equation is equal to zero.
Finding the Discriminant
The discriminant of a quadratic equation of the form \( ax^2 + bx + c = 0 \) is given by the formula:
\[ \Delta = b^2 - 4ac \]
For the given equation \( x^2 - mx + 4 = 0 \), we have \( a = 1 \), \( b = -m \), and \( c = 4 \).
Substitute these values into the formula to find the discriminant:
\[ \Delta = (-m)^2 - 4(1)(4) = m^2 - 16 \]
Equal Roots Condition
For a quadratic equation to have real and equal roots, the discriminant must be equal to zero.
Therefore, we set \( \Delta = 0 \) and solve for \( m \):
\[ m^2 - 16 = 0 \]
\[ m^2 = 16 \]
\[ m = \pm 4 \]
Conclusion
The value of \( m \) for which the quadratic equation \( x^2 - mx + 4 = 0 \) has real and equal roots is \( m = \pm 4 \).
Therefore, the correct answer is option (D) \( m = -4 \).
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For what value of m will the quadratic equation x2 - mx + 4 = 0 have r...
Step 1: Nature of Roots of Quadratic Equations Theory
D is the Discriminant in a quadratic equation.
D = b2 - 4ac for quadratic equations of the form ax2 + bx + c = 0.
If D > 0, roots are Real and Unique (Distinct and real roots).
If D = 0, roots are Real and Equal.
If D < 0, roots are Imaginary. The roots of such quadratic equations are NOT real.
The quadratic equation given in this question has real and equal roots. Therefore, its discriminant D = 0.
D = b2 - 4ac for quadratic equations of the form ax2 + bx + c = 0.
If D > 0, roots are Real and Unique (Distinct and real roots).
If D = 0, roots are Real and Equal.
If D < 0, roots are Imaginary. The roots of such quadratic equations are NOT real.
The quadratic equation given in this question has real and equal roots. Therefore, its discriminant D = 0.
Step 2: Compute discriminant for the equation in terms of ‘m’ and find the value of ‘m’.
In the given equation x2 - mx + 4 = 0, a = 1, b = -m and c = 4.
Therefore, the discriminant b2 - 4ac = m2 - 4(4)(1) = m2 - 16.
The roots of the given equation are real and equal.
Therefore, m2 - 16 = 0 or m2 = 16 or m = +4 or m = -4.
Therefore, the discriminant b2 - 4ac = m2 - 4(4)(1) = m2 - 16.
The roots of the given equation are real and equal.
Therefore, m2 - 16 = 0 or m2 = 16 or m = +4 or m = -4.
Choice (D) is the answer to this quadratic equations question.
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