the sum of two roots of the equation...

[chick7007]

the sum of the two roots of the equation 2x^2-ax+b=X is 7. what is the value of constant a?

i think the answer is 14. but my answer sheet says it's 13.......
i want to know who is correct

this is my work
(2x-8)(x-3)
4+3=7
(2x-12)(x-1)
6+1=7
therefore a=14

....i'm not sure if this guess and check process was the most efficient and correct way to solve this problem...
please help me

 

[HallsofIvy]

If the problem had been "\(\displaystyle 2x^2+ ax+ b= 0\)", then you would have been correct. But it wasn't. It was, according to you, "\(\displaystyle 2x^2+ ax+ b= x\)". You forgot to allow for the "x" on the right. \(\displaystyle 2x^2+ 14x+ b= x\) is the same as \(\displaystyle 2x^2+ 13x+ b= 0\).

 

[daon2]

You might want to remember:

If \(\displaystyle f(x)=Ax^2+Bx+C = A(x-\alpha)(x-\beta)\) Then the sum of the roots of \(\displaystyle f(x)\) is

\(\displaystyle \alpha+\beta = -\dfrac{B}{A}\)

Geometrically, this is a result of the axis of symmetry (when the roots are real). The roots are centered at \(\displaystyle x = -\dfrac{B}{2A}\) so when adding them we get \(\displaystyle 2\left(\dfrac{-B}{2A}\right) = -\dfrac{B}{A}\)

 

[Deleted member 4993]

Another good relationship to remember:

\(\displaystyle \alpha \ * \ \beta \ = \ \dfrac{C}{A}\)