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Solve this 3rd equation in english

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Answer

−516-\frac{5}{16}

Step-by-step solution

4 steps

  1. Compare the given polynomial f(x)=4x2−5x−1f(x) = 4x^2 - 5x - 1 with the standard quadratic polynomial ax2+bx+cax^2 + bx + c.

    Here, a=4a=4, b=−5b=-5, and c=−1c=-1.

  2. Find the sum of the roots α+β\alpha + \beta.

    The sum of the roots is given by the formula: α+β=−ba\alpha + \beta = -\frac{b}{a}

    α+β=−(−5)4=54\alpha + \beta = -\frac{(-5)}{4} = \frac{5}{4}

  3. Find the product of the roots αβ\alpha \beta.

    The product of the roots is given by the formula: αβ=ca\alpha \beta = \frac{c}{a}

    αβ=−14\alpha \beta = \frac{-1}{4}

  4. Find the value of α2β+αβ2\alpha^2\beta + \alpha\beta^2.

    First, factor out the common term αβ\alpha\beta from the expression:

    α2β+αβ2=αβ(α+β)\alpha^2\beta + \alpha\beta^2 = \alpha\beta(\alpha + \beta)

    Now, substitute the values obtained from Step 2 and Step 3:

    α2β+αβ2=(−14)(54)\alpha^2\beta + \alpha\beta^2 = \left(-\frac{1}{4}\right) \left(\frac{5}{4}\right) α2β+αβ2=−1×54×4=−516\alpha^2\beta + \alpha\beta^2 = -\frac{1 \times 5}{4 \times 4} = -\frac{5}{16}

    The value of α2β+αβ2\alpha^2\beta + \alpha\beta^2 is −516-\frac{5}{16}.

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