contestada

Neil and Chris were trying to solve the quadratic equation
\[x^2 + bx + c = 0.\]Neil wrote down the wrong value of $b$ (but his value of $c$ was correct), and found the roots to be $1$ and $6.$ Chris wrote down the wrong value of $c$ (but his value of $b$ was correct), and found the roots to be $-1$ and $-4.$ What are the actual roots of $x^2 + bx + c = 0$?

Respuesta :

Let [tex]x_1,\ x_2[/tex] be the roots of the equation [tex]x^2+bx+c=0[/tex].

The coefficients and the roots are in the following relation:

[tex]x_1+x_2=-b,\quad x_1x_2=c[/tex]

So, since Neil got the correct value of [tex]c[/tex], we know that the product of his roots is correct: the product of the roots must be 6.

Similarly, since Chris got the correct value of [tex]b[/tex], we know that the opposite of the sum of his roots is correct: the opposite of the sum of the roots must be 5.

So, we want two numbers that give -5 when summed, and 6 when multiplied. Those numbers are -2 and -3.

Otras preguntas