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

In summary, to have two real roots, the discriminant of a quadratic equation must be greater than 0. Using the quadratic formula, we can find the values of a that satisfy this condition, which are a < (-3 - √7) / 2 or a > (-3 + √7) / 2. To determine if a is between x1 and x2, we can calculate x1 and x2 using the quadratic formula.
  • #1
songoku
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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) / 2Then how to know if a is between x1 and x2? Thanks
 
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  • #2
songoku said:
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) / 2Then how to know if a is between x1 and x2? Thanks
Why not calculate ##x_1## and##x_2##?
 
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Likes songoku
  • #3
PeroK said:
Why not calculate ##x_1## and##x_2##?
I understand. Thank you very much PeroK
 

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

1. What is the meaning of "a" in a quadratic equation?

"a" represents the coefficient of the quadratic term in a quadratic equation, which determines the shape and direction of the parabola.

2. How does "a" affect the two real roots of a quadratic equation?

The value of "a" determines the position of the roots on the x-axis. If "a" is positive, the parabola opens upwards and the roots will be on opposite sides of the vertex. If "a" is negative, the parabola opens downwards and the roots will be on the same side of the vertex.

3. Can "a" be equal to 0 in a quadratic equation?

No, "a" cannot be equal to 0 in a quadratic equation as it would result in a linear equation, which only has one real root.

4. How can the value of "a" be determined from the two real roots of a quadratic equation?

The value of "a" can be determined using the quadratic formula, which is (-b ± √(b²-4ac)) / 2a. By plugging in the values of the two real roots and solving for "a", the coefficient can be found.

5. What is the significance of the value of "a" in a quadratic equation?

The value of "a" is important in understanding the behavior of a quadratic equation and its graph. It affects the position of the vertex, the direction of the parabola, and the number and position of the roots.

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