Question

Please help me with this problem:Consider the quadratic equation 3x^2 - 6 = 2x.(a)What is the value of the discriminant?(b)What does the discriminant of the quadratic equation tell about the solutions to 3x^2 - 6 = 2x ?

Answer 1

Consider the general quadratic equation:

`ax^2+bx+c=0`

The discriminant of this equation is defined as:

`d=b^2-4ac`

(a) Calculate the discriminant. Before we can use the formula, we need to transform the equation into the required form.

We are given:

`3x^2-6=2x`

Subtracting 2x and rearranging:

`3x^2-2x-6=0`

We can now identify the coefficients: a = 3, b = -2, c = -6, and compute the discriminant as follows:

`\begin{gathered} d=(-2)^2-4*3*(-6) \\ d=4+72 \\ \boxed{d=76} \end{gathered}`

(b) The discriminant gives important information about the solutions of the equation:

* If d is zero, there is only one real solution.

* If d is positive, there are two real solutions.

* If d is negative, there are two complex solutions.

In our equation, d is positive and the equation has two real solutions