Question

Use the discriminant to describe the root of each equation 3x^2-10=0​

Answer 1

Given a quadratic equation
`ax^2+bx+c=0`

The discriminant is defined as

`\Delta=b^2-4ac`

In your case, the equation is defined by the coefficients

`a=3,b=0,c=-10`

So, the discriminant is

`\Delta=0^2-4\cdot 3\cdot (-10) = 120`

The discriminant is involved in the solving formula as follows:

`x_(1,2) = (-b\pm√(\Delta))/(2a)`

Which implies that:

• If
  `\Delta>0` the root exists, and so you have two distinct solution (the one where you choose the plus sign, and the one where you choose the minus sign in the solving formula)
• If
  `\Delta=0` the root is zero, and you have two collpapsed solutions, since there's no difference in adding or subtracting it.
• If
  `\Delta<0`, the root is not defined using real numbers, and the equation has no real solutions.

In your case, since the discriminant is positive, you have two distinct solutions. Since 120 is not a perfect square, however, you will not get rid of the square root, so you will have two distinct irrational solutions.