Answer:
The sum of the roots of the equation [tex]x^{2} + x = 2[/tex] is -1
Step-by-step explanation:
You have two options to find the sum of the roots,
[tex]x_{1,2} = \frac{-b\±\sqrt{b^{2}-4ac}}{2a} [/tex]
[tex]x^{2} + x - 2= [/tex] where:
a = 1
b = 1
c = -2
[tex]x_{1} = \frac{-1-\sqrt{1^{2}-4*1*-2}}{2*1}[/tex] = -2
[tex]x_{2} = \frac{-1+\sqrt{1^{2}-4*1*-2}}{2*1}[/tex] = 1
The sum of the roots is -2 + 1 = -1
2. The second option is use the fact that a general quadratic equation is in the form of:
[tex]ax^{2}+bx+c=0[/tex]
if you divided by [tex]a[/tex] you get:
[tex]x^{2}+\frac{b}{a} x+\frac{c}{a} =0[/tex]
and always the sum of roots will be given for this expression [tex]x_{1} + x_{2} = \frac{-b}{a}[/tex]
Why this is true?
Because if we use the Quadratic Formula as follows:
[tex]x_{1} + x_{2} = \frac{-b+\sqrt{b^{2}-4ac}}{2*a} + \frac{-b-\sqrt{b^{2}-4ac}}{2a}[/tex]
[tex]x_{1} + x_{2} = \frac{-2b+0}{2a}}[/tex]
[tex]x_{1} + x_{2} = \frac{-b}{a}[/tex]
In the case of this equation:
[tex]x_{1} + x_{2} = \frac{-1}{1} = -1[/tex]
