Answer:
x = 3
b = 42
Step-by-step explanation:
The root of a quadratic equation is a value for the variable that makes the equation equal to zero.
If one of the roots of the quadratic equation 6x² - bx + 72 = 0 is equal to 4, it means that when we substitute x = 4 into the equation, the expression evaluates to zero. Therefore, to find the value of b, substitute x = 4 into the equation:
[tex]\begin{aligned}6(4)^2-b(4)+72&=0\\6(16)-4b+72&=0\\96-4b+72&=0\\-4b+168&=0\\4b&=168\\b&=42\end{aligned}[/tex]
So, the quadratic equation is 6x² - 42x + 72 = 0.
According to the Factor Theorem, if 'c' is a root of a polynomial function, then (x - c) is a factor of the polynomial. Therefore, if one of the roots of the quadratic equation is 4, then one of its factors is (x - 4). To find the other factor, we can divide the quadratic by (x - 4):
[tex]\large \begin{array}{r}6x-18\phantom{)}\\x-4{\overline{\smash{\big)}\,6x^2-42x+72\phantom{)}}\\{-~\phantom{(}\underline{(6x^2-24x)\phantom{-b.b)}}\\-18x+72\phantom{)}\\-~\phantom{()}\underline{(-18x+72)\phantom{}}\\0\phantom{)}\\\end{array}[/tex]
Therefore, the other factor is (6x - 18).
To find the other root, set the factor to zero and solve for x:
[tex]\begin{aligned}6x-18&=0\\6x&=18\\x&=3\end{aligned}[/tex]
So, the other root is x = 3.
