Question

Find the values of k for which the equation (k – 1)x^2 + kx – k = 0 has real and
distinct roots.

Answer 1

☯︎ Aɴsᴡᴇʀ

we have a quadratic equation

• (k -1)x² + kx - k = 0
• equation has real and distinct roots

firstly compare the equation with ax² + bx + c = 0

then here,

• a = k - 1
• b = k
• c = -k

_____________________

☯︎ first we have to know the concept,

The quadratic equation ax² + bx + c = 0 has

• two distinct real roots, if D = b² - 4ac > 0
• two equal real roots, it D = b² - 4ac = 0
• no real roots, if D = b² - 4ac < 0

D = b² - 4ac

D = discriminate of the quadratic equation

_____________________

and it is given that, roots of the quadratic equation (k - 1)x² + Kx - k = 0 has real and distinct roots.

➪ D > 0

➪ b² - 4ac > 0

➪ k² - 4 × (k - 1) × (-k) > 0

➪ k²- (4k - 4) × (-k) > 0

➪ k² - (-4k² + 4k) > 0

➪ k² + 4k² + 4k > 0

➪ k(k + 4k + 4) > 0

➪ k + 4k + 4 > 0

➪ 5k + 4 > 0

➪ 5k > - 4

➪ k > -4/5

so, value of k is greater than -4/5