If the roots of the equation x2−ax+b=0 satisfy the equation 2x2−(a+3)x+(3b−2)=0, then....
- a=3
- b=2
- 2a−2ab+3b=0
- ab=5
Explanation
Since both quadratic equations have the same roots, their coefficients can be equated.
2x2−(a+3)x+(3b−2)=2(x2−ax+b)
2x2−(a+3)x+(3b−2)=2x2−2ax+2b
Testing the first statement
Equate the coefficient of x from both equations.
−(a+3)=−2a
−a−3=−2a
a=3
The first statement is correct.
Testing the second statement
Equate the constant from both equations.
3b−2=2b
b=2
The second statement is correct.
Testing the third statement
Substitute the values a=3 and b=2 into the third statement.
2a−2ab+3b=2(3)−2(3)(2)+3(2)
2a−2ab+3b=6−12+6
2a−2ab+3b=0
The third statement is correct.
Testing the fourth statement
Substitute the values a=3 and b=2 into the fourth statement.
ab=(2)(3)=6
The fourth statement is incorrect because the result is not 5.
Therefore, the correct statements are the first, second, and third.