Value of "a" between two real roots of quadratic equation

[songoku]

Homework Statement

The equation 2x^2 - 2(2a + 1)x + a(a - 1) = 0 has two real roots x1 and x2. Find a such that x1 < a < x2 !

Relevant Equations

Discriminant
Quadratic formula

For quadratic equation to have two real roots:

b² - 4ac > 0

(-2 (2a + 1))² - 4 (2) (a (a - 1)) > 0

4 (4a² + 4a + 1) - 8a² + 8a > 0

16a² + 16a + 4 - 8a² + 8a > 0

8a² + 24 a + 4 > 0

2a² + 6a + 1 > 0

Using quadratic formula, I get a < (-3 - √7) / 2 or a > (-3 + √7) / 2


Then how to know if a is between x₁ and x₂? Thanks

 

[PeroK]

Homework Statement:: The equation 2x^2 - 2(2a + 1)x + a(a - 1) = 0 has two real roots x1 and x2. Find a such that x1 < a < x2 !
Relevant Equations:: Discriminant
Quadratic formula

For quadratic equation to have two real roots:

b² - 4ac > 0

(-2 (2a + 1))² - 4 (2) (a (a - 1)) > 0

4 (4a² + 4a + 1) - 8a² + 8a > 0

16a² + 16a + 4 - 8a² + 8a > 0

8a² + 24 a + 4 > 0

2a² + 6a + 1 > 0

Using quadratic formula, I get a < (-3 - √7) / 2 or a > (-3 + √7) / 2


Then how to know if a is between x₁ and x₂? Thanks

Why not calculate $x_1$ and$x_2$?

 

[songoku]

Why not calculate $x_1$ and$x_2$?

I understand. Thank you very much PeroK