To solve the quadratic equation [tex]\( y^2 - a y - 6 = 0 \)[/tex] for [tex]\( y \)[/tex] in terms of the constant [tex]\( a \)[/tex], follow these steps:
1. Identify the coefficients: Compare the given equation to the standard form of a quadratic equation [tex]\( ay^2 + by + c = 0 \)[/tex]. Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -a \)[/tex], and [tex]\( c = -6 \)[/tex].
2. Write down the quadratic formula: The solutions to the quadratic equation [tex]\( ay^2 + by + c = 0 \)[/tex] are given by:
[tex]\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
3. Substitute the coefficients into the quadratic formula: For our equation [tex]\( y^2 - a y - 6 = 0 \)[/tex], we have:
[tex]\[
y = \frac{-(-a) \pm \sqrt{(-a)^2 - 4(1)(-6)}}{2(1)}
\][/tex]
4. Simplify inside the square root:
[tex]\[
y = \frac{a \pm \sqrt{a^2 - 4(1)(-6)}}{2}
\][/tex]
[tex]\[
y = \frac{a \pm \sqrt{a^2 + 24}}{2}
\][/tex]
5. Write the final solutions: Therefore, the solutions for [tex]\( y \)[/tex] in terms of [tex]\( a \)[/tex] are:
[tex]\[
y = \frac{a - \sqrt{a^2 + 24}}{2} \quad \text{and} \quad y = \frac{a + \sqrt{a^2 + 24}}{2}
\][/tex]
So, the expression for [tex]\( y \)[/tex] in terms of [tex]\( a \)[/tex] are:
[tex]\[
y = \frac{a - \sqrt{a^2 + 24}}{2} \quad \text{and} \quad y = \frac{a + \sqrt{a^2 + 24}}{2}
\][/tex]
