Chapter 1.9, Problem 11Q
After evaluating the discriminant of the quadratic equation 0=−x2−4x−4 , state the number and types of solution of it.
The discriminant of the quadratic equation is 0 and it has a unique real solution x=−2 .
Given:
The quadratic equation is 0=−x2−4x−4 which can be written as x2+4x+4=0 .
Concept used:
Any quadratic equation of the form ax2+bx+c=0 where a≠0 has the roots x=−b±D2a ......1
And the term D=b2−4ac is called discriminant D of the quadratic equation. Now to find nature and number of roots of quadratic equation three cases arises.
1 If discriminant D>0 , then the quadratic equation has two real and distinct solutions.
2 If discriminant D<0 , then the quadratic equation has two imaginary distinct solutions which will be complex conjugate of each other.
3 If discriminant D=0 , then the quadratic equation a unique real solution.
Calculation:
Evaluate discriminant D of the quadratic equation x2+4x+4=0 .
D=42−414 ......x2+4x+4=0=0
Now discriminant D=0 , therefore the quadratic equation has a unique real solution using clause 3 in concept state above.
Now, the solution for the quadratic equation x2+4x+4=0 can be evaluated with the help of formula 1 . Therefore,
x=−b±D2a ......From 1=−4±021 ......∵x2+4x+4=0 and D=0=−42=−2
Conclusion:
The discriminant of the quadratic equation is 0 and it has a unique real solution x=−2 .