Find m, if x^2+mx+25=0 has equal roots
Answers
Answer
The required value of m is 10
Given
The quadratic equation is :
- x² + mx + 25 = 0
- The given has real and equal roots
To Find
- The value of m
Solution
We are given that the equation
x² + mx + 25 = 0 has equal roots
Therefore , its discriminant b² - 4ac will be equal to 0
Here in the equation,
a = 1
b = m
c = 25
∴ b² - 4ac = 0
⇒ m² - 4×1×25 = 0
⇒ m² - 100 = 0
⇒ m² = 100
⇒ m² = 10²
⇒ m = 10
Thus , the value of m is 10
Answer:
m = ± 10
Explanation:
ATQ, x² + mx + 25 = 0 has equal roots.
We know that if a quadratic equation has equal roots, its discriminant will be 0.
⇒ Discriminant (D) = 0
⇒ b² - 4ac = 0
Where:
b → Coefficient of x
a → Coefficient of x²
c → Constant term.
Given Polynomial:
p(x) = x² + mx + 25
Discriminant = 0
⇒ b² - 4ac = 0
⇒ (m)² - 4(1)(25) = 0
⇒ m² - 100 = 0
⇒ m² = 100
⇒ m = √100
⇒ m = ± 10
Since we have two possible answers, we'll substitute it in p(x) to check if both answers are valid.
Case 1:
When m = + 10
p(x) = x² + mx + 25
p(x) = x² + 10x + 25
p(x) = x² + 5x + 5x + 25
p(x) = x(x + 5) + 5(x + 5)
p(x) = (x + 5) (x + 5)
x = -5 & -5.
The roots are equal, therefore, This case is valid.
Case 2:
When m = - 10
p(x) = x² + mx + 25
p(x) = x² + (-10)x + 25
p(x) = x² - 10x + 25
p(x) = x² - 5x - 5x + 25
p(x) = x(x - 5) - 5(x - 5)
p(x) = (x - 5) (x - 5)
x = 5 & 5.
The roots are equal, therefore, This case is valid too.