QuestionJuly 17, 2025

Solve the equation 6y^3-7y^2-3y=0 y=square Write your answers as a list of integers or reduced fractions, with your answers separated by (a) comma(s). For example, if you get 4 and -(2)/(3) as your answers, then enter 4,-2/3 in the box.

Solve the equation 6y^3-7y^2-3y=0 y=square Write your answers as a list of integers or reduced fractions, with your answers separated by (a) comma(s). For example, if you get 4 and -(2)/(3) as your answers, then enter 4,-2/3 in the box.
Solve the equation 6y^3-7y^2-3y=0 y=square Write your answers as a list of integers or reduced fractions, with your answers separated by (a) comma(s). For example, if you get 4 and -(2)/(3) as your answers, then enter 4,-2/3 in the box.

Solution
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Answer

0, \frac{3}{2}, -\frac{1}{3} Explanation 1. Factor out the common term The equation is \( 6y^3 - 7y^2 - 3y = 0 \). Factor out ( y ): \[ y(6y^2 - 7y - 3) = 0 \] 2. Solve for the first root Set ( y = 0 ), which gives one solution: \[ y = 0 \] 3. Factor the quadratic Factor \( 6y^2 - 7y - 3 \) using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where ( a = 6, b = -7, c = -3 ): \[ y = \frac{-(-7) \pm \sqrt{(-7)^2 - 4 \cdot 6 \cdot (-3)}}{2 \cdot 6} \] 4. Calculate the discriminant Compute the discriminant: \[ b^2 - 4ac = 49 + 72 = 121 \] 5. Solve for remaining roots Substitute back into the quadratic formula: \[ y = \frac{7 \pm \sqrt{121}}{12} \] Simplify: \[ y = \frac{7 \pm 11}{12} \] 6. Find specific solutions Calculate the two possible values: \[ y = \frac{18}{12} = \frac{3}{2} \] \[ y = \frac{-4}{12} = -\frac{1}{3} \]

Explanation

1. Factor out the common term<br /> The equation is \( 6y^3 - 7y^2 - 3y = 0 \). Factor out ( y ): <br />\[ y(6y^2 - 7y - 3) = 0 \]<br /><br />2. Solve for the first root<br /> Set ( y = 0 ), which gives one solution:<br />\[ y = 0 \]<br /><br />3. Factor the quadratic<br /> Factor \( 6y^2 - 7y - 3 \) using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where ( a = 6, b = -7, c = -3 ):<br />\[ y = \frac{-(-7) \pm \sqrt{(-7)^2 - 4 \cdot 6 \cdot (-3)}}{2 \cdot 6} \]<br /><br />4. Calculate the discriminant<br /> Compute the discriminant:<br />\[ b^2 - 4ac = 49 + 72 = 121 \]<br /><br />5. Solve for remaining roots<br /> Substitute back into the quadratic formula:<br />\[ y = \frac{7 \pm \sqrt{121}}{12} \]<br /> Simplify:<br />\[ y = \frac{7 \pm 11}{12} \]<br /><br />6. Find specific solutions<br /> Calculate the two possible values:<br />\[ y = \frac{18}{12} = \frac{3}{2} \]<br />\[ y = \frac{-4}{12} = -\frac{1}{3} \]
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