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Sagot :
To solve the equation [tex]\(\frac{11y - 2}{3} = y^2 + 2\)[/tex], we need to eliminate the fraction and solve for [tex]\(y\)[/tex]. Follow these steps:
1. Eliminate the fraction: Multiply both sides of the equation by 3 to clear the denominator.
[tex]\[ 3 \cdot \frac{11y - 2}{3} = 3 \cdot (y^2 + 2) \][/tex]
This simplifies to:
[tex]\[ 11y - 2 = 3y^2 + 6 \][/tex]
2. Rearrange the equation: Move all terms to one side to set the equation to zero.
[tex]\[ 11y - 2 - 3y^2 - 6 = 0 \][/tex]
Simplifying this, we get:
[tex]\[ -3y^2 + 11y - 8 = 0 \][/tex]
or equivalently:
[tex]\[ 3y^2 - 11y + 8 = 0 \][/tex]
3. Solve the quadratic equation: We use the quadratic formula [tex]\( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex] where [tex]\( a = 3 \)[/tex], [tex]\( b = -11 \)[/tex], and [tex]\( c = 8 \)[/tex].
Calculate the discriminant:
[tex]\[ b^2 - 4ac = (-11)^2 - 4(3)(8) \][/tex]
[tex]\[ = 121 - 96 \][/tex]
[tex]\[ = 25 \][/tex]
Since the discriminant is positive, we have two distinct real solutions. Now, find the roots:
[tex]\[ y = \frac{-(-11) \pm \sqrt{25}}{2 \cdot 3} \][/tex]
[tex]\[ y = \frac{11 \pm 5}{6} \][/tex]
4. Calculate the roots:
[tex]\[ y_1 = \frac{11 + 5}{6} = \frac{16}{6} = \frac{8}{3} \][/tex]
[tex]\[ y_2 = \frac{11 - 5}{6} = \frac{6}{6} = 1 \][/tex]
Hence, the solutions to the equation [tex]\(\frac{11y - 2}{3} = y^2 + 2\)[/tex] are:
[tex]\[ y = 1 \quad \text{and} \quad y = \frac{8}{3} \][/tex]
1. Eliminate the fraction: Multiply both sides of the equation by 3 to clear the denominator.
[tex]\[ 3 \cdot \frac{11y - 2}{3} = 3 \cdot (y^2 + 2) \][/tex]
This simplifies to:
[tex]\[ 11y - 2 = 3y^2 + 6 \][/tex]
2. Rearrange the equation: Move all terms to one side to set the equation to zero.
[tex]\[ 11y - 2 - 3y^2 - 6 = 0 \][/tex]
Simplifying this, we get:
[tex]\[ -3y^2 + 11y - 8 = 0 \][/tex]
or equivalently:
[tex]\[ 3y^2 - 11y + 8 = 0 \][/tex]
3. Solve the quadratic equation: We use the quadratic formula [tex]\( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex] where [tex]\( a = 3 \)[/tex], [tex]\( b = -11 \)[/tex], and [tex]\( c = 8 \)[/tex].
Calculate the discriminant:
[tex]\[ b^2 - 4ac = (-11)^2 - 4(3)(8) \][/tex]
[tex]\[ = 121 - 96 \][/tex]
[tex]\[ = 25 \][/tex]
Since the discriminant is positive, we have two distinct real solutions. Now, find the roots:
[tex]\[ y = \frac{-(-11) \pm \sqrt{25}}{2 \cdot 3} \][/tex]
[tex]\[ y = \frac{11 \pm 5}{6} \][/tex]
4. Calculate the roots:
[tex]\[ y_1 = \frac{11 + 5}{6} = \frac{16}{6} = \frac{8}{3} \][/tex]
[tex]\[ y_2 = \frac{11 - 5}{6} = \frac{6}{6} = 1 \][/tex]
Hence, the solutions to the equation [tex]\(\frac{11y - 2}{3} = y^2 + 2\)[/tex] are:
[tex]\[ y = 1 \quad \text{and} \quad y = \frac{8}{3} \][/tex]
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