Explanation

y = 7x - 5x\(^2\)

\(\frac{dy}{dx}\) = slope/gradient

\(\frac{dy}{dx}\) = 7 - 10x  at x = 2

m\(_1\) = - 13

But,the equation of normal, we need m\(_2\), from m\(_1\) m\(_2\) = - 1

m\(_2\) = \(\frac{-1}{m_1}\) = \(\frac{-1}{-13}\) = \(\frac{1}{13}\)

Using, \(\frac{ y - y_1}{x - x_1}\) = \(\frac{1}{m_2}\)

\(\frac{ y - y_1}{x - x_1}\) = \(\frac{1}{m_2}\)

To find the value of y we put x = 2 into y = 7x - 5x\(^2\) = 14 - 20 = -6

\(\frac{ y - (-6)}{x - 2}\) = \(\frac{1}{m_2}\)

y + 6 = \(\frac{1}{m_2}\)(x - 2)

13y + 78 = x - 2

13y - x + 80 = 0

Thus, the equation of normal = 13y - x + 80 = 0

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