If x1 and x2 are the roots of x^2 + 3x - 5 = 0, find the value of x1 + x2 - x1*x2.

Step-by-step solution

1. 1

   Recognise a symmetric expression in the roots. $x_1 + x_2 - x_1x_2$ is built only from the sum and the product of the roots, so Vieta's formulas answer it directly — solving the quadratic is unnecessary and would drag irrational surds through the arithmetic.

2. 2

   Write down Vieta's formulas. For $ax^2 + bx + c = 0$,

   $x_1 + x_2 = -\frac{b}{a}, \qquad x_1x_2 = \frac{c}{a}$

3. 3

   Substitute the coefficients. Here $a = 1$, $b = 3$, $c = -5$:

   $x_1 + x_2 = -\frac{3}{1} = -3, \qquad x_1x_2 = \frac{-5}{1} = -5$

4. 4

   Combine.

   $x_1 + x_2 - x_1x_2 = -3 - (-5) = -3 + 5 = 2$

   Subtracting a negative product is where a sign slip would turn the answer into $-8$.

5. 5

   Verify with the actual roots. The discriminant is $9 + 20 = 29$, so $x_{1,2} = \tfrac{-3 \pm \sqrt{29}}{2}$, numerically $1.192582$ and $-4.192582$. Their sum is $-3.000000$ and their product is $-5.000000$, giving $-3 + 5 = 2$ as expected.