Find the zeros for each of the following quadratics. If you use the quadratic formula please report \( D \) and write out your formulas. Please round decimal answers to two decimal places. 1. \( x^{2}-12 x+32 \) Solve the following equations. 3. \( x^{2}+12 x=-11 \) 3. Thes of quccircitic \( 6 x^{2}-7 x=14 \)

To find the zeros for the quadratic \( x^{2}-12x+32 \), we can first calculate the discriminant \( D \). \( D = b^2 - 4ac = (-12)^2 - 4(1)(32) = 144 - 128 = 16 \). Now we apply the quadratic formula \( x = \frac{-b \pm \sqrt{D}}{2a} \): \( x = \frac{12 \pm \sqrt{16}}{2(1)} = \frac{12 \pm 4}{2} \). Thus, the solutions are \( x = \frac{16}{2} = 8 \) and \( x = \frac{8}{2} = 4 \). For the equation \( x^{2}+12x=-11 \), we first rearrange it to standard form: \( x^{2}+12x+11 = 0 \). Here, \( D = 12^2 - 4(1)(11) = 144 - 44 = 100 \). Using the quadratic formula again: \( x = \frac{-12 \pm \sqrt{100}}{2(1)} = \frac{-12 \pm 10}{2} \). So, the solutions are \( x = \frac{-2}{2} = -1 \) and \( x = \frac{-22}{2} = -11 \). For the equation \( 6x^{2}-7x=14 \), we rearrange to: \( 6x^{2}-7x-14 = 0 \). Calculating the discriminant gives: \( D = (-7)^2 - 4(6)(-14) = 49 + 336 = 385 \). Now, applying the quadratic formula: \( x = \frac{7 \pm \sqrt{385}}{2(6)} \). This results in: \( x = \frac{7 \pm 19.62}{12} \). Hence, the solutions are approximately \( x = \frac{26.62}{12} \approx 2.22 \) and \( x = \frac{-12.62}{12} \approx -1.05 \). Solving these quadratics not only helps in understanding their properties but also provides skills useful in various real-world scenarios, like optimization problems in engineering and economics where you might need to find maximum or minimum values. Additionally, common mistakes in solving quadratics often stem from mishandling the discriminant or applying the quadratic formula incorrectly. Double-checking signs and ensuring proper order of operations can save a lot of headaches!