Set the polynomial to zero and solve: \( x^{2} - 9x + 14 = 0 \)

To solve the quadratic equation \( x^{2} - 9x + 14 = 0 \), you can use several methods such as factoring, completing the square, or the quadratic formula. Here, we'll demonstrate the **factoring method** and the **quadratic formula**. ### Method 1: Factoring 1. **Write the equation in standard form:** \[ x^{2} - 9x + 14 = 0 \] 2. **Find two numbers that multiply to **+14** and add to **-9**:** - The numbers **-7** and **-2** satisfy these conditions because: \[ (-7) \times (-2) = 14 \quad \text{and} \quad (-7) + (-2) = -9 \] 3. **Factor the quadratic using these numbers:** \[ x^{2} - 7x - 2x + 14 = 0 \\ (x^{2} - 7x) + (-2x + 14) = 0 \\ x(x - 7) - 2(x - 7) = 0 \\ (x - 7)(x - 2) = 0 \] 4. **Set each factor equal to zero and solve for \( x \):** \[ x - 7 = 0 \quad \Rightarrow \quad x = 7 \\ x - 2 = 0 \quad \Rightarrow \quad x = 2 \] ### Method 2: Quadratic Formula The quadratic formula states that for any quadratic equation \( ax^{2} + bx + c = 0 \), the solutions for \( x \) are: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] 1. **Identify the coefficients \( a \), \( b \), and \( c \):** \[ a = 1, \quad b = -9, \quad c = 14 \] 2. **Plug the coefficients into the quadratic formula:** \[ x = \frac{-(-9) \pm \sqrt{(-9)^{2} - 4(1)(14)}}{2(1)} \\ x = \frac{9 \pm \sqrt{81 - 56}}{2} \\ x = \frac{9 \pm \sqrt{25}}{2} \\ x = \frac{9 \pm 5}{2} \] 3. **Calculate the two possible solutions:** \[ x = \frac{9 + 5}{2} = \frac{14}{2} = 7 \\ x = \frac{9 - 5}{2} = \frac{4}{2} = 2 \] ### **Solution** Both methods yield the same solutions. Therefore, the solutions to the equation \( x^{2} - 9x + 14 = 0 \) are: \[ \boxed{x = 2 \quad \text{and} \quad x = 7} \]