Finding the Common Root of Two Quadratic Equations

We need to find the value of '$x$' that satisfies both equations:

1. $x^2 - 21x + 110 = 0$
2. $110 + x - x^2 = 0$

Solving the First Equation

Consider the first equation: $x^2 - 21x + 110 = 0$. We look for two numbers that multiply to 110 and add up to -21. These numbers are -10 and -11.

Factor the equation:

$(x - 10)(x - 11) = 0$

The roots are:

$x = 10$ or $x = 11$

Solving the Second Equation

Consider the second equation: $110 + x - x^2 = 0$. Rearrange it into standard form:

$-x^2 + x + 110 = 0$

Multiply the entire equation by -1 to make the $x^2$ term positive:

$x^2 - x - 110 = 0$

We look for two numbers that multiply to -110 and add up to -1. These numbers are -11 and +10.

Factor the equation:

$(x - 11)(x + 10) = 0$

The roots are:

$x = 11$ or $x = -10$

Identifying the Common Root

Compare the roots from both equations:

• Roots of Equation 1: {10, 11}
• Roots of Equation 2: {-10, 11}

The value that appears in both sets of roots is 11.

Conclusion

The common root of the two equations is 11.