Questions: Find all the zeros of the quadratic function. y=x^2-11x-26

Solution Steps

To find the zeros of the quadratic function \( y = x^2 - 11x - 26 \), we need to solve the equation \( x^2 - 11x - 26 = 0 \). This can be done using 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 \( ax^2 + bx + c = 0 \).

1. Identify the coefficients \( a = 1 \), \( b = -11 \), and \( c = -26 \).
2. Use the quadratic formula to calculate the roots of the equation.
3. Compute the discriminant \( b^2 - 4ac \) to determine the nature of the roots.
4. Calculate the two possible values for \( x \) using the quadratic formula.

The given quadratic function is \( y = x^2 - 11x - 26 \). The coefficients are:

• \( a = 1 \)
• \( b = -11 \)
• \( c = -26 \)

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( \Delta = b^2 - 4ac \). \[ \Delta = (-11)^2 - 4 \times 1 \times (-26) = 121 + 104 = 225 \]

The roots of the quadratic equation are given by: \[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \] Substituting the values: \[ x_1 = \frac{-(-11) + \sqrt{225}}{2 \times 1} = \frac{11 + 15}{2} = \frac{26}{2} = 13.0 \] \[ x_2 = \frac{-(-11) - \sqrt{225}}{2 \times 1} = \frac{11 - 15}{2} = \frac{-4}{2} = -2.0 \]

Final Answer