Question

Use the discriminant to describe the roots of each equation. Then select the best description.

7x² + 1 = 5x

Answer 1

Imaginary roots

Explanation:

The discriminant of a quadratic in standard form
`ax^2+bx+c` is given by
`b^2-4ac`.

Given
`7x^2+1=5x`, subtract 5x from both sides so that the quadratic is in standard form:

`7x^2-5x+1=0`

Now assign variables:

• 
  `a\implies 7`
• 
  `b\implies -5`
• 
  `c\implies 1`

The discriminant is therefore
`(-5)^2-4(7)(1)=25-28=\textbf{-3}`.

What does this tell us about the roots?

Recall that the discriminant is what is under the radical in the quadratic formula. The quadratic formula is used to find the solutions of a quadratic. Therefore, the solutions of this quadratic would be equal to
`(-b\pm √(-3))/(2a)` for some
`b` and
`a`. Since the number under the radical is negative, there are no real roots to the quadratic (whenever the discriminant is negative, the are zero real solutions to the quadratic). Therefore, the quadratic has imaginary roots.