Question

What is the discriminant of `5x^2-8x-3=0` and what does that mean?

Answer 1

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=124`

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;

`5x^2−8x−3=0`

Now compare the above equation with quadratic equation `ax^2+bx+c=0`, we get `a=5, b=-8 and c = -3`.

Hence the discriminant (D) is given by;

`D = b^2-4ac`
`=> D = (-8)^2 - 4*5*(-3)`
`=> D = 64-(-60)`
`=> D = 64+60=124`

Therefore the discriminant of a given equation is 124.

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.

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