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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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Verified Answer
For what value of 'm' will the quadratic equation x2- mx + 4 =...
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.
Choice (D) is the answer to this quadratic equations question.
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.
Choice (D) is the answer to this quadratic equations question.
Most Upvoted Answer
For what value of 'm' will the quadratic equation x2- mx + 4 =...
Understanding Quadratic Equations
A quadratic equation in the form ax² + bx + c = 0 has real and equal roots when its discriminant (D) is equal to zero. The discriminant is calculated using the formula:
D = b² - 4ac
In this case, the equation is x² - mx + 4 = 0. Here, a = 1, b = -m, and c = 4.
Finding the Discriminant
To find the condition for real and equal roots:
- Substitute a, b, and c into the discriminant formula:
D = (-m)² - 4(1)(4)
- This simplifies to:
D = m² - 16
Setting the Discriminant to Zero
For the roots to be real and equal, we set the discriminant equal to zero:
m² - 16 = 0
Solving for m
Now, solve for m:
- Rearranging gives:
m² = 16
- Taking the square root of both sides results in:
m = ±4
Identifying the Correct Option
From the values obtained (m = 4 and m = -4), we can see that:
- The possible values of m are 4 and -4.
- The question specifies the correct option as 'D', which corresponds to -4.
Therefore, the value of m that will ensure the quadratic equation x² - mx + 4 = 0 has real and equal roots is indeed -4.
A quadratic equation in the form ax² + bx + c = 0 has real and equal roots when its discriminant (D) is equal to zero. The discriminant is calculated using the formula:
D = b² - 4ac
In this case, the equation is x² - mx + 4 = 0. Here, a = 1, b = -m, and c = 4.
Finding the Discriminant
To find the condition for real and equal roots:
- Substitute a, b, and c into the discriminant formula:
D = (-m)² - 4(1)(4)
- This simplifies to:
D = m² - 16
Setting the Discriminant to Zero
For the roots to be real and equal, we set the discriminant equal to zero:
m² - 16 = 0
Solving for m
Now, solve for m:
- Rearranging gives:
m² = 16
- Taking the square root of both sides results in:
m = ±4
Identifying the Correct Option
From the values obtained (m = 4 and m = -4), we can see that:
- The possible values of m are 4 and -4.
- The question specifies the correct option as 'D', which corresponds to -4.
Therefore, the value of m that will ensure the quadratic equation x² - mx + 4 = 0 has real and equal roots is indeed -4.
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