Question

Tell how many solutions the equation has. $$ 6 x^{2}-12 x-6=0 $$

Short answer

The given equation has two real and distinct solutions.

Step-by-step solution

Identify a, b, and c from the given quadratic equation

The equation \(6x^{2}-12x-6=0\) is in the form \(ax^{2}+bx+c=0\). So, here \(a=6\), \(b=-12\), and \(c=-6\).

Substitute a, b, and c into the formula for the discriminant

The discriminant (\(\Delta\)) is given by the formula \(b^{2}-4ac\). Substituting the values of \(a\), \(b\), and \(c\) from Step 1 into this formula: \(\Delta = (-12)^{2} - 4 * 6 * -6\).

Calculate the discriminant

Compute the value of \(\Delta = 144 - (-144) = 288\).

Check the nature of solutions

Since our calculated discriminant (\(\Delta = 288\)) is greater than 0, it means that the given quadratic equation has two real and distinct solutions.