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The roots of 7x^2 + x - 5 = 0 are a and b. Compute (a - 4) + (b - 4).

 Jun 16, 2022
 #1
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This question has already been answered

 

https://web2.0calc.com/questions/help-please_52172#r3

 

-Vinculum

 

smiley

 Jun 16, 2022
 #2
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Alternative solution:

 

Since we know the roots are a and b, we know that \(7x^2 + x - 5 = 7(x - a)(x - b)\).

 

Expanding the right-hand side gives \(7x^2 + x - 5 = 7x^2 - 7(a + b)x + 7ab\).

 

Comparing the coefficients of x on both sides: \(1 = -7(a + b)\).

 

Therefore, \(a + b = -\dfrac17\).

 

Now, \((a - 4) + (b - 4) = (a + b) - 8\), using this gives \((a - 4) + (b - 4) = -\dfrac17 - 8 =\boxed{ -\dfrac{57}8}\).

 Jun 16, 2022

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