Answer:
-3/5.
Step-by-step explanation:
2x² + 3x + 5
α + β = b/a = -3/2
αβ = c/a = 5/2
So, 1/α + 1/β
= (α + β) / αβ
= -3/2 / 5/2
= -3/2 * 2/5
= -6/10
= -3/5.
[tex]{\huge{\bf{\pink{†{\red{\underline{\orange{\mathfrak{Añßwér:}}}}}}}}}[/tex]
[tex]{\color{lime}{\underline{\rule{100pt}{2pt}}}}{\color{magenta}{\underline{\rule{100pt}{2pt}}}} [/tex]
↦
In this case,
↦[tex]\rm{a = 2}[/tex],
↦[tex]\sf{b = 3}[/tex],
↦[tex]\rm{c = 5}[/tex].
[tex]\sf{\alpha = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 2 \cdot 5}}{2 \cdot 2}}[/tex]
Simplifying:
↦[tex]\rm{\bold{\green{\alpha = \frac{-3 \pm \sqrt{9 - 40}}{4}}}}[/tex]
↦[tex]\sf{\bold{\orange{\alpha = \frac{-3 \pm \sqrt{-31}}{4}}}}[/tex]
Since the quadratic has complex roots,
↦[tex]\rm{\alpha}[/tex] ,
↦[tex]\sf{\beta}[/tex]
, ↦[tex]\sf{\beta}[/tex] can be written as
↦ [tex]\overline{\alpha}[/tex] (the conjugate of [tex]\sf{\alpha}[/tex]
let's find the value of
↦ [tex]\sf{\frac{1}{\alpha} + \frac{1}{\beta}}[/tex].
↦[tex]\rm{\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta}{\alpha \cdot \beta} + \frac{\alpha}{\alpha \cdot \beta}}[/tex]
Since:
↦ [tex]\rm{\bold{\beta = \overline{\alpha}}}[/tex],
↦[tex]\sf{\frac{1}{\alpha} + \frac{1}{\overline{\alpha}} = \frac{\overline{\alpha}}{\alpha \cdot \overline{\alpha}} + \frac{\alpha}{\alpha \cdot \overline{\alpha}}}[/tex]
Simplifying, we get:
↦[tex]\rm{\frac{\overline{\alpha} + \alpha}{\alpha \cdot \overline{\alpha}}}[/tex]
Using the property that
↦[tex]\rm{\alpha \cdot \overline{\alpha}}[/tex] is equal to the square of the absolute value of [tex]\sf{\alpha}[/tex], we can rewrite the equation:
↦[tex]\rm{\frac{\overline{\alpha} + \alpha}{|\alpha|^2}}[/tex]
Since : [tex]\alpha[/tex] and [tex]\sf{\beta}[/tex] are complex conjugates,
↦ [tex]\rm{\overline{\alpha} = \beta}[/tex].
↦[tex]\sf{\frac{\beta + \alpha}{|\alpha|^2}}[/tex]
Therefore,
↦[tex]\rm{\bold{\orange{\boxed{\underline{\underline{\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{|\alpha|^2}}}}}}}[/tex].
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Answers & Comments
Answer:
-3/5.
Step-by-step explanation:
2x² + 3x + 5
α + β = b/a = -3/2
αβ = c/a = 5/2
So, 1/α + 1/β
= (α + β) / αβ
= -3/2 / 5/2
= -3/2 * 2/5
= -6/10
= -3/5.
Step-by-step explanation:
[tex]{\huge{\bf{\pink{†{\red{\underline{\orange{\mathfrak{Añßwér:}}}}}}}}}[/tex]
[tex]{\color{lime}{\underline{\rule{100pt}{2pt}}}}{\color{magenta}{\underline{\rule{100pt}{2pt}}}} [/tex]
↦
In this case,
↦[tex]\rm{a = 2}[/tex],
↦[tex]\sf{b = 3}[/tex],
↦[tex]\rm{c = 5}[/tex].
[tex]\sf{\alpha = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 2 \cdot 5}}{2 \cdot 2}}[/tex]
Simplifying:
↦[tex]\rm{\bold{\green{\alpha = \frac{-3 \pm \sqrt{9 - 40}}{4}}}}[/tex]
↦[tex]\sf{\bold{\orange{\alpha = \frac{-3 \pm \sqrt{-31}}{4}}}}[/tex]
Since the quadratic has complex roots,
↦[tex]\rm{\alpha}[/tex] ,
↦[tex]\sf{\beta}[/tex]
, ↦[tex]\sf{\beta}[/tex] can be written as
↦ [tex]\overline{\alpha}[/tex] (the conjugate of [tex]\sf{\alpha}[/tex]
let's find the value of
↦ [tex]\sf{\frac{1}{\alpha} + \frac{1}{\beta}}[/tex].
↦[tex]\rm{\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta}{\alpha \cdot \beta} + \frac{\alpha}{\alpha \cdot \beta}}[/tex]
Since:
↦ [tex]\rm{\bold{\beta = \overline{\alpha}}}[/tex],
↦[tex]\sf{\frac{1}{\alpha} + \frac{1}{\overline{\alpha}} = \frac{\overline{\alpha}}{\alpha \cdot \overline{\alpha}} + \frac{\alpha}{\alpha \cdot \overline{\alpha}}}[/tex]
Simplifying, we get:
↦[tex]\rm{\frac{\overline{\alpha} + \alpha}{\alpha \cdot \overline{\alpha}}}[/tex]
Using the property that
↦[tex]\rm{\alpha \cdot \overline{\alpha}}[/tex] is equal to the square of the absolute value of [tex]\sf{\alpha}[/tex], we can rewrite the equation:
↦[tex]\rm{\frac{\overline{\alpha} + \alpha}{|\alpha|^2}}[/tex]
Since : [tex]\alpha[/tex] and [tex]\sf{\beta}[/tex] are complex conjugates,
↦ [tex]\rm{\overline{\alpha} = \beta}[/tex].
↦[tex]\sf{\frac{\beta + \alpha}{|\alpha|^2}}[/tex]
Therefore,
↦[tex]\rm{\bold{\orange{\boxed{\underline{\underline{\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{|\alpha|^2}}}}}}}[/tex].
[tex]{\color{lime}{\underline{\rule{100pt}{2pt}}}}{\color{magenta}{\underline{\rule{100pt}{2pt}}}} [/tex]