Question

x^2 − 5mx + 6m^2 − 4m − 16 = 0.
a) Aflați m ∈ R dacă x = m este soluție a ecuației;
b) Rezolvați ecuația dată.

Answer 1

$\displaystyle{ x^{2} - 5mx + 6m^{2} - 4m - 16 = 0 }$

$\displaystyle{ x = m \rightarrow ecuatia \ devine \ x^{2} - 5x \cdot x + 6x^{2} - 4x - 16 = 0 }$

$\displaystyle{ x^{2} - 5x^{2} + 6x^{2} - 4x - 16 = 0 }$

$\displaystyle{ -4x^{2} + 6x^{2} - 4x - 16 = 0 }$

$\displaystyle{ 2x^{2} - 4x - 16 = 0 }$

• a = 2
• b = -4
• c = -16

$\displaystyle{ \Delta = b^{2} - 4 \cdot a \cdot c = (-4)^{2} - 4 \cdot 2 \cdot (-16) }$

$\displaystyle{ \Delta = 16 + 8 \cdot 16 = 16 +128 = 144 }$

$\displaystyle{ x_{1} = \frac{-b + \sqrt{\Delta}}{2a} = \frac{4+\sqrt{122}}{2\cdot 2} }$

$\displaystyle{ x_{1} = \frac{4 + 12}{4} = \frac{16}{4} = 4 }$

$\displaystyle{ x_{2} = \frac{-b-\sqrt{\Delta}}{2a} = \frac{4-\sqrt{122}}{2 \cdot 2} }$

$\displaystyle{ x_{2} = \frac{4-12}{4} = \frac{-8}{4} = -2 }$

m ∈ {-2, 4}