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

1 Answer
Jul 24, 2015

Answer:

The discriminant of an equation tells the nature of the roots of a quadratic equation given that a,b and c are rational numbers.

#D=48#

Explanation:

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