Question

without solving the given equation find the value of 'm' for which the given equation has real and equal roots. x² + 2(m -1)x + (m+5) = 0

Answer 1

Answer

The value of m is -1 or 4

Given

The quadratic equation is

• x² + 2(m - 1)x + (m + 5) = 0
• Also the roots of the given equation are real and equal

To Find

• The value of m

Concept to be used

Discriminant of a quadratic equation :

If ax² + bx + c is a quadratic equation then its discriminant is given by D = b² - 4ac

We can predict the nature of roots from discriminant i.e.

• If b² - 4ac = 0 then the roots are real and equal
• If b² - 4ac ≥ 0 then the roots are real
• If b² - 4ac ≤ 0 then real roots does not exist ( immaginary roots exist)

Solution

We are given the equation is

• x² + 2(m - 1)x + (m + 5)

Since the roots are real and equal so D = 0 i.e.

⇒{2(m - 1)}² - 4×1×(m + 5) = 0

⇒4(m - 1)² - 4(m + 5) = 0

⇒ 4(m² - 2m + 1 - m - 5 )=0

⇒ m² - 3m - 4 = 0

⇒ m² - 4m + m - 4 = 0

⇒ m(m - 4) + 1(m - 4) = 0

⇒ (m + 1)(m - 4) = 0

Thus ,

⇒ m + 1 = 0 and ⇒m - 4= 0

⇒ m = -1 and ⇒m = 4