+1791 Let $a$ and $b$ be the roots of $x^2 + 7x - 4 = -x^2 + 9x + 6$. Find $(a + 3) + (b + 3).$
+16 x2 + x2 + 7x - 9x - 4 - 6 = 0
2x2 - 2x - 10 = 0
Divide both sides by 2:
x2 - x - 5 = 0
Now, find the roots using the quadratic formula:
x = 1 +/- √1 - 4(1)(-5) 1 +/- √21
------------------------ = -----------------
2 2
So, a = (1 + √21) / 2, b = (1 - √21) / 2
We are trying to find what is (a + 3) + (b + 3), so we just plug in the numbers:
1 + √21 1 - √21 1 √21 1 √21
( ----------- + 3) + ( ---------- + 3) = 3 -- + ----- + 3 --- - ----- = 3.5 + 3.5 = 7
2 2 2 2 2 2
