To solve the quadratic equation [tex]\( y^2 - 9y + 14 = 0 \)[/tex], we can use the quadratic formula, which is given by:
[tex]\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
In this quadratic equation, [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are the coefficients of [tex]\(y^2\)[/tex], [tex]\(y\)[/tex], and the constant term, respectively. Here, [tex]\(a = 1\)[/tex], [tex]\(b = -9\)[/tex], and [tex]\(c = 14\)[/tex].
First, we need to calculate the discriminant, which is [tex]\( b^2 - 4ac \)[/tex].
1. Calculate the discriminant:
[tex]\[ \Delta = (-9)^2 - 4 \cdot 1 \cdot 14 \][/tex]
[tex]\[ \Delta = 81 - 56 \][/tex]
[tex]\[ \Delta = 25 \][/tex]
Since the discriminant is positive, we have two real solutions. Now, we use the quadratic formula to find these solutions.
2. Calculate the solutions:
[tex]\[ y_1 = \frac{-(-9) + \sqrt{25}}{2 \cdot 1} \][/tex]
[tex]\[ y_1 = \frac{9 + 5}{2} \][/tex]
[tex]\[ y_1 = \frac{14}{2} \][/tex]
[tex]\[ y_1 = 7 \][/tex]
[tex]\[ y_2 = \frac{-(-9) - \sqrt{25}}{2 \cdot 1} \][/tex]
[tex]\[ y_2 = \frac{9 - 5}{2} \][/tex]
[tex]\[ y_2 = \frac{4}{2} \][/tex]
[tex]\[ y_2 = 2 \][/tex]
Thus, the solutions to the equation [tex]\( y^2 - 9y + 14 = 0 \)[/tex] are:
[tex]\[ y = 7, 2 \][/tex]
