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Détermine la longueur et la largeur d'un rectangle de périmètre 60 cm et d'aire 221 cm².

Mathematics

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Answer

13(x−6)(x+3)\frac{1}{3}(x-6)(x+3)

Step-by-step solution

4 steps

  1. Identify the coefficients of the quadratic function f(x)=13x2−x−6f(x) = \frac{1}{3}x^2 - x - 6.

    The general form of a quadratic function is ax2+bx+cax^2 + bx + c.
    Here, a=13a = \frac{1}{3}, b=−1b = -1, and c=−6c = -6.

  2. Calculate the discriminant Δ\Delta using the formula Δ=b2−4ac\Delta = b^2 - 4ac.

    Δ=(−1)2−4(13)(−6)\Delta = (-1)^2 - 4 \left(\frac{1}{3}\right) (-6) Δ=1−4(−2)\Delta = 1 - 4 \left(-2\right) Δ=1+8\Delta = 1 + 8 Δ=9\Delta = 9

  3. Find the roots of the quadratic equation using the formula x=−b±Δ2ax = \frac{-b \pm \sqrt{\Delta}}{2a}.

    Since Δ=9>0\Delta = 9 > 0, there are two distinct real roots.

    x=−(−1)±92(13)x = \frac{-(-1) \pm \sqrt{9}}{2 \left(\frac{1}{3}\right)} x=1±323x = \frac{1 \pm 3}{\frac{2}{3}}

    Calculate the two roots:

    x1=1+323=423=4×32=6x_1 = \frac{1 + 3}{\frac{2}{3}} = \frac{4}{\frac{2}{3}} = 4 \times \frac{3}{2} = 6 x2=1−323=−223=−2×32=−3x_2 = \frac{1 - 3}{\frac{2}{3}} = \frac{-2}{\frac{2}{3}} = -2 \times \frac{3}{2} = -3

  4. Factorize the polynomial using the roots.

    The factorization of a quadratic ax2+bx+cax^2 + bx + c is a(x−x1)(x−x2)a(x-x_1)(x-x_2).

    f(x)=13(x−6)(x−(−3))f(x) = \frac{1}{3}(x-6)(x-(-3)) f(x)=13(x−6)(x+3)f(x) = \frac{1}{3}(x-6)(x+3)

    The factorization of f(x)f(x) is:

    13(x−6)(x+3)\frac{1}{3}(x-6)(x+3)
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