We know that the discriminant for the quadratic equation `ax^2+bx+c=0` is `D=b^2-4ac.`
Now we have the following conditions based on the values of `D` :
i) If `D>0` both roots of the quadratic equation are real and distinct.
ii) If `D=0` both roots of the quadratic equation are real and equal.
iii) If `D<0` both roots of the quadratic equation are complex numbers.
Now given problem is `-3x^2-7x+8=10` . Which can be written as
Here `a=-3, b=-7, c=-2.`
So, by condition (i) both roots of the given quadratic equation are real and distinct.
Our equation can be written as
`3x^2+7x+2=0` , multiplying both sides by (-1).
Now `(3x+1)=0rArr x=-1/3` ,
and `(x+2)=0rArr x=-2` .
So, `x=-1/3, -2.`
The discriminant of quadratic equation ax^2 + bx +c = 0 is b^2-4ac.
But before applying this formula, set one side of the given equation equal to zero. To do so, subtract both sides by 10.
Then, plug-in the values of a, b and c to the formula of discriminant. The values are a=-3, b=-7 and c=-2.
Hence, the discriminant is 25.
Also, since the value of discriminant is greater tha zero and a perfect square (25=5^2), therefore, the solutions of the given equation are two rational numbers.
take away 10
`a=-3` `b=-7` `c=-2` use `b^2-4ac`
`49-24=25 ` the discriminant is 25 and since it is bigger than 0 it means it has 2 real solutions. The roots can be found using the quadratic equation.
`(7+5)/(-6) = (12)/(-6) = -2`
`(7-5)/(-6) = 2/(-6) = 1/-3`
the roots are -2 and 1/-3