What is the discriminant of `4x^2-64x+145=-8x-3` and what does that mean?

The discriminant of a quadratic equation `ax^2+bx+c=0` is given by the formula `b^2+4ac` of the quadratic formula;

`x = (-b+-sqrt{b^2-4ac})/(2a)`

The discriminant actually tells you the nature of the roots of a quadratic equation or in other words, the number of x-intercepts, associated with a quadratic equation.

Now we have an equation;

`4x^2−64x+145=−8x−3`

First transform it to standard form of the quadratic equation.

`4x^2−64x+145+8x+3=0` `=>` Added `8x` and `3` on both side.
or, `4x^2-56x+148=0 =>` Combined like terms.
or, `x^2-14x+37=0 =>` Divided both side by 4.

Now compare the above equation with quadratic equation `ax^2+bx+c=0`, we get `a=1, b=-14 and c = 37`.

Hence the discriminant (D) is given by;

`D = b^2-4ac`
`=> D = (-14)^2 - 4*1*37`
`=> D = 196-148`
`=> D = 48`

Therefore the discriminant of a given equation is 48.

Here the discriminant is greater than 0 i.e. `b^2-4ac>0`, hence there are two real roots.

Note: If the discriminant is a perfect square, the two roots are rational numbers. If the discriminant is not a perfect square, the two roots are irrational numbers containing a radical.

Thanks