Questions: n(v)=-x^2+5x+1

Solution Steps

To find the roots of the quadratic function \( n(v) = -x^2 + 5x + 1 \), we can use the quadratic formula, which is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

where \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \). In this case, \( a = -1 \), \( b = 5 \), and \( c = 1 \).

For the quadratic function \( n(v) = -x^2 + 5x + 1 \), we identify the coefficients as follows:

• \( a = -1 \)
• \( b = 5 \)
• \( c = 1 \)

The discriminant \( D \) is calculated using the formula: \[ D = b^2 - 4ac \] Substituting the values: \[ D = 5^2 - 4(-1)(1) = 25 + 4 = 29 \] Taking the square root gives: \[ \sqrt{D} = \sqrt{29} \approx 5.3852 \]

Using the quadratic formula: \[ x = \frac{-b \pm \sqrt{D}}{2a} \] we find the two roots: \[ x_1 = \frac{-5 + \sqrt{29}}{2(-1)} = \frac{-5 + 5.3852}{-2} \approx -0.1926 \] \[ x_2 = \frac{-5 - \sqrt{29}}{2(-1)} = \frac{-5 - 5.3852}{-2} \approx 5.1926 \]

Final Answer