Which of the following quadratic equations has equal roots?
- Ax
2
−14x−49=0 - Bx
2
+7x+49=0 - Cx
2
−7x−49=0 - Dx
2
+14x+49=0
Solution & Step-by-step Explanation
A quadratic equation ax
2
+bx+c=0 has equal roots if its discriminant (D) is equal to zero:
D=b
2
−4ac=0
Let's evaluate the options:
Option A: x
2
−14x−49=0
D=(−14)
2
−4(1)(−49)=196+196=392
=0
Option B: x
2
+7x+49=0
D=(7)
2
−4(1)(49)=49−196=−147
=0
Option C: x
2
−7x−49=0
D=(−7)
2
−4(1)(−49)=49+196=245
=0
Option D: x
2
+14x+49=0
D=(14)
2
−4(1)(49)=196−196=0
Since D=0 for Option D, this equation has equal roots. Alternatively, it can be written as a perfect square: (x+7)
2
=0.
2
+bx+c=0 has equal roots if its discriminant (D) is equal to zero:
D=b
2
−4ac=0
Let's evaluate the options:
Option A: x
2
−14x−49=0
D=(−14)
2
−4(1)(−49)=196+196=392
=0
Option B: x
2
+7x+49=0
D=(7)
2
−4(1)(49)=49−196=−147
=0
Option C: x
2
−7x−49=0
D=(−7)
2
−4(1)(−49)=49+196=245
=0
Option D: x
2
+14x+49=0
D=(14)
2
−4(1)(49)=196−196=0
Since D=0 for Option D, this equation has equal roots. Alternatively, it can be written as a perfect square: (x+7)
2
=0.