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