# What are the zero(s) 3x^2-7x+12=0?

Nov 8, 2015

$3 {x}^{2} - 7 x + 12 = 0$ has no zeros

#### Explanation:

For a parabolic equation in the form
$\textcolor{w h i t e}{\text{XXX}} a {x}^{2} + b x + c = 0$
the discriminant
color(white)("XXX) Delta = b^2-4ac
indicates the number of zeros for the equation.

Specifically, in this case when
$\textcolor{w h i t e}{\text{XXX}} \Delta < 0$
there are no solutions (i.e. no zeros)

For the given equation, you can see in the graph below that expression $3 {x}^{2} - 7 x + 12$ never touches the X-axis (i.e. it is never equal to zero).
graph{3x^2-7x+12 [-13.75, 26.8, -2.68, 17.59]}
The discriminant is part of the quadratic formula which gives the solutions for equations of this type:
$\textcolor{w h i t e}{\text{XXX}} x = \frac{- b \pm \sqrt{\textcolor{b l u e}{{b}^{2} - 4 a c}}}{2 a}$
as you can see if the discriminant is zero then the solution would require the square root of a negative number
and the square root of a negative number does not exist as a Real value.