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In the quadratic equation, 6x^2-gx+2=0 , the sum of the roots is equal to three times their product

Sagot :

Semsee45idn

Answer:

g = 6

Step-by-step explanation:

To find the value of g in quadratic equation 6x² - gx + 2 = 0, we can use Vieta's formulas.

[tex]\boxed{\begin{array}{l}\underline{\textsf{Vieta's Formulas for Quadratic Equation}}\\\\\textsf{If $f(x) = ax^2+ bx + c$ has roots $\alpha$ and $\beta$ then:}\\\\\textsf{Sum of the roots:}\quad \alpha+\beta=-\dfrac{b}{a}\\\\\textsf{Product of the roots}\quad \alpha\cdot\beta=\dfrac{c}{a}\end{array}}[/tex]

Given that the sum of the roots is equal to three times their product:

[tex]-\dfrac{b}{a}=3\cdot \dfrac{c}{a}[/tex]

In this case:

  • a = 6
  • b = -g
  • c = 2

Substitute these values into the equation:

[tex]-\dfrac{-g}{6}=3\cdot \dfrac{2}{6}[/tex]

Now, solve for g:

[tex]\dfrac{g}{6}=3\cdot \dfrac{2}{6}\\\\\\\dfrac{g}{6}=\dfrac{6}{6}\\\\\\\dfrac{g}{6}=1\\\\\\g=6[/tex]

Therefore, the value of g is:

[tex]\LARGE\boxed{\boxed{g=6}}[/tex]

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