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3y2 − 5y + 2 = 0

Mathematics

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Answer

y=1,23y = 1, \frac{2}{3}

Step-by-step solution

6 steps

To solve the quadratic equation 3y2−5y+2=03y^2 - 5y + 2 = 0, we can use the factoring method.

  1. Identify the coefficients of the quadratic equation.

    The equation is in the form ay2+by+c=0ay^2 + by + c = 0, where a=3a=3, b=−5b=-5, and c=2c=2.

  2. Find two numbers that multiply to acac and add up to bb.

    We need two numbers that multiply to 3×2=63 \times 2 = 6 and add up to −5-5. These numbers are −2-2 and −3-3.

  3. Rewrite the middle term using these two numbers.

    3y2−2y−3y+2=03y^2 - 2y - 3y + 2 = 0

  4. Factor by grouping.

    Group the terms and factor out common factors from each group:

    (3y2−2y)+(−3y+2)=0(3y^2 - 2y) + (-3y + 2) = 0 y(3y−2)−1(3y−2)=0y(3y - 2) - 1(3y - 2) = 0

  5. Factor out the common binomial term.

    (3y−2)(y−1)=0(3y - 2)(y - 1) = 0

  6. Set each factor equal to zero and solve for yy.

    For the first factor:

    3y−2=03y - 2 = 0 3y=23y = 2 y=23y = \frac{2}{3}

    For the second factor:

    y−1=0y - 1 = 0 y=1y = 1

    The solutions for yy are:

    y=1,23y = 1, \frac{2}{3}
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