Question

При каких значениях параметра а уравнение 4х^2 + 3ах + 1 = 0 имеет два различных корня?

Answer 1

При:
$> x∈<img src=$∪$(\frac{4}{3};\ +\infty )$

Answer 2

Finding the Values of Parameter "a" for the Equation to Have Two Distinct Roots

To find the values of the parameter "a" for the equation \(4x^2 + 3ax + 1 = 0\) to have two distinct roots, we can use the discriminant of the quadratic equation.

The discriminant, denoted as \(D\), is given by the formula: \[D = b^2 - 4ac\] where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation \(ax^2 + bx + c = 0\).

For the equation \(4x^2 + 3ax + 1 = 0\), the discriminant is: \[D = (3a)^2 - 4*4*1\]

To have two distinct roots, the discriminant \(D\) must be greater than 0.

Calculating the Discriminant for Two Distinct Roots

To find the values of "a" for which the equation has two distinct roots, we set the discriminant \(D\) greater than 0 and solve for "a": \[D > 0\] \[(3a)^2 - 16 > 0\]

Solving the inequality: \[(3a)^2 > 16\] \[9a^2 > 16\] \[a^2 > \frac{16}{9}\] \[a > \pm \frac{4}{3}\]

Therefore, the equation \(4x^2 + 3ax + 1 = 0\) has two distinct roots when the parameter "a" is greater than or less than \(\pm \frac{4}{3}\).

In summary, the equation \(4x^2 + 3ax + 1 = 0\) has two distinct roots when the parameter "a" is greater than or less than \(\pm \frac{4}{3}\).