Question

find the sum and the product of the roots of each equation.
$x^2 - 37x + 27 = 0$
answer
the sum is \\(\square\\), the product is \\(\square\\).

Explanation:

Step1: Recall Vieta's formulas for quadratic equation

For a quadratic equation of the form \(ax^{2}+bx + c = 0\) (where \(a
eq0\)), if the roots are \(r_1\) and \(r_2\), then the sum of the roots \(r_1 + r_2=-\frac{b}{a}\) and the product of the roots \(r_1\times r_2=\frac{c}{a}\).

Step2: Identify \(a\), \(b\), \(c\) from the given equation

Given the equation \(x^{2}-37x + 27 = 0\), we can see that \(a = 1\), \(b=- 37\), and \(c = 27\).

Step3: Calculate the sum of the roots

Using the formula for the sum of the roots \(r_1 + r_2=-\frac{b}{a}\), substitute \(a = 1\) and \(b=-37\) into the formula:
\(r_1 + r_2=-\frac{-37}{1}=37\)

Step4: Calculate the product of the roots

Using the formula for the product of the roots \(r_1\times r_2=\frac{c}{a}\), substitute \(a = 1\) and \(c = 27\) into the formula:
\(r_1\times r_2=\frac{27}{1}=27\)

Answer:

The sum is \(37\), the product is \(27\).