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If x(x-3)=-1 then the value of x^(3)(x^(3)-18) will be,

If x(x−3)=−1 x(x-3)=-1 then the value of x3(x3−18) x^{3}\left(x^{3}-18\right) will be,

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Full solution

Q. If x(x−3)=−1 x(x-3)=-1 then the value of x3(x3−18) x^{3}\left(x^{3}-18\right) will be,
  1. Given Equation: We are given the equation x(x−3)=−1x(x-3) = -1.\newlineTo find the value of x3(x3−18)x^{3}(x^{3}-18), we first need to find the value of xx.\newlineLet's solve the given equation for xx.\newlinex2−3x+1=0x^2 - 3x + 1 = 0\newlineThis is a quadratic equation, and we can solve it using the quadratic formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where a=1a = 1, b=−3b = -3, and c=1c = 1.
  2. Quadratic Formula: Calculate the discriminant b2−4acb^2 - 4ac of the quadratic equation.\newlineDiscriminant = (−3)2−4(1)(1)(-3)^2 - 4(1)(1)\newlineDiscriminant = 9−49 - 4\newlineDiscriminant = 55\newlineThe discriminant is positive, so we have two real and distinct solutions for xx.
  3. Discriminant Calculation: Apply the quadratic formula to find the two solutions for xx.x=−(−3)±5(2⋅1)x = \frac{-(-3) \pm \sqrt{5}}{(2 \cdot 1)}x=3±52x = \frac{3 \pm \sqrt{5}}{2}So, the two solutions for xx are 3+52\frac{3 + \sqrt{5}}{2} and 3−52\frac{3 - \sqrt{5}}{2}.
  4. Solving Quadratic Equation: Now we need to find the value of x3(x3−18)x^{3}(x^{3}-18) for each solution of xx. Let's first consider x=3+52x = \frac{3 + \sqrt{5}}{2}. We will calculate x3x^3 and then use it to find x3(x3−18)x^{3}(x^{3}-18). x3=(3+52)3x^3 = \left(\frac{3 + \sqrt{5}}{2}\right)^3 This involves expanding the cube of a binomial.
  5. First Solution Calculation: Expand the cube of the binomial (3+5)/2(3 + \sqrt{5})/2.
    x3=[(3+5)/2]3x^3 = [(3 + \sqrt{5})/2]^3
    x3=(3/2+5/2)3x^3 = (3/2 + \sqrt{5}/2)^3
    x3=(3/2)3+3∗(3/2)2∗(5/2)+3∗(3/2)∗(5/2)2+(5/2)3x^3 = (3/2)^3 + 3*(3/2)^2*(\sqrt{5}/2) + 3*(3/2)*(\sqrt{5}/2)^2 + (\sqrt{5}/2)^3
    This step involves binomial expansion and simplification.
  6. Second Solution Calculation: Simplify the expression for x3x^3.
    x3=278+3(94)(52)+3(32)(54)+558x^3 = \frac{27}{8} + 3\left(\frac{9}{4}\right)\left(\frac{\sqrt{5}}{2}\right) + 3\left(\frac{3}{2}\right)\left(\frac{5}{4}\right) + \frac{5\sqrt{5}}{8}
    x3=278+27516+4516+558x^3 = \frac{27}{8} + \frac{27\sqrt{5}}{16} + \frac{45}{16} + \frac{5\sqrt{5}}{8}
    x3=(27+4516)+(275+10516)x^3 = \left(\frac{27 + 45}{16}\right) + \left(\frac{27\sqrt{5} + 10\sqrt{5}}{16}\right)
    x3=7216+37516x^3 = \frac{72}{16} + \frac{37\sqrt{5}}{16}
    x3=92+37516x^3 = \frac{9}{2} + \frac{37\sqrt{5}}{16}
    Now we have the value of x3x^3 for the first solution.
  7. Correcting Approach: Repeat the process for the second solution x=3−52x = \frac{3 - \sqrt{5}}{2}.x3=(3−52)3x^3 = \left(\frac{3 - \sqrt{5}}{2}\right)^3This involves expanding the cube of a binomial, similar to the previous steps.
  8. Correcting Approach: Repeat the process for the second solution x=3−52x = \frac{3 - \sqrt{5}}{2}.
    x3=[3−52]3x^3 = \left[\frac{3 - \sqrt{5}}{2}\right]^3
    This involves expanding the cube of a binomial, similar to the previous steps.Expand the cube of the binomial 3−52\frac{3 - \sqrt{5}}{2}.
    x3=[3−52]3x^3 = \left[\frac{3 - \sqrt{5}}{2}\right]^3
    x3=(32−52)3x^3 = \left(\frac{3}{2} - \frac{\sqrt{5}}{2}\right)^3
    x3=(32)3−3(32)2(52)+3(32)(52)2−(52)3x^3 = \left(\frac{3}{2}\right)^3 - 3\left(\frac{3}{2}\right)^2\left(\frac{\sqrt{5}}{2}\right) + 3\left(\frac{3}{2}\right)\left(\frac{\sqrt{5}}{2}\right)^2 - \left(\frac{\sqrt{5}}{2}\right)^3
    This step involves binomial expansion and simplification.
  9. Correcting Approach: Repeat the process for the second solution x=3−52x = \frac{3 - \sqrt{5}}{2}.
    x3=[3−52]3x^3 = \left[\frac{3 - \sqrt{5}}{2}\right]^3
    This involves expanding the cube of a binomial, similar to the previous steps.Expand the cube of the binomial 3−52\frac{3 - \sqrt{5}}{2}.
    x3=[3−52]3x^3 = \left[\frac{3 - \sqrt{5}}{2}\right]^3
    x3=(32−52)3x^3 = \left(\frac{3}{2} - \frac{\sqrt{5}}{2}\right)^3
    x3=(32)3−3(32)2(52)+3(32)(52)2−(52)3x^3 = \left(\frac{3}{2}\right)^3 - 3\left(\frac{3}{2}\right)^2\left(\frac{\sqrt{5}}{2}\right) + 3\left(\frac{3}{2}\right)\left(\frac{\sqrt{5}}{2}\right)^2 - \left(\frac{\sqrt{5}}{2}\right)^3
    This step involves binomial expansion and simplification.Simplify the expression for x3x^3.
    x3=278−3(94)(52)+3(32)(54)−558x^3 = \frac{27}{8} - 3\left(\frac{9}{4}\right)\left(\frac{\sqrt{5}}{2}\right) + 3\left(\frac{3}{2}\right)\left(\frac{5}{4}\right) - 5\frac{\sqrt{5}}{8}
    x3=278−27516+4516−558x^3 = \frac{27}{8} - \frac{27\sqrt{5}}{16} + \frac{45}{16} - \frac{5\sqrt{5}}{8}
    x3=27+4516−275+10516x^3 = \frac{27 + 45}{16} - \frac{27\sqrt{5} + 10\sqrt{5}}{16}
    x3=[3−52]3x^3 = \left[\frac{3 - \sqrt{5}}{2}\right]^300
    x3=[3−52]3x^3 = \left[\frac{3 - \sqrt{5}}{2}\right]^311
    Now we have the value of x3x^3 for the second solution.
  10. Correcting Approach: Repeat the process for the second solution x=3−52x = \frac{3 - \sqrt{5}}{2}.
    x3=(3−52)3x^3 = \left(\frac{3 - \sqrt{5}}{2}\right)^3
    This involves expanding the cube of a binomial, similar to the previous steps.Expand the cube of the binomial 3−52\frac{3 - \sqrt{5}}{2}.
    x3=(3−52)3x^3 = \left(\frac{3 - \sqrt{5}}{2}\right)^3
    x3=(32−52)3x^3 = \left(\frac{3}{2} - \frac{\sqrt{5}}{2}\right)^3
    x3=(32)3−3(32)2(52)+3(32)(52)2−(52)3x^3 = \left(\frac{3}{2}\right)^3 - 3\left(\frac{3}{2}\right)^2\left(\frac{\sqrt{5}}{2}\right) + 3\left(\frac{3}{2}\right)\left(\frac{\sqrt{5}}{2}\right)^2 - \left(\frac{\sqrt{5}}{2}\right)^3
    This step involves binomial expansion and simplification.Simplify the expression for x3x^3.
    x3=278−3(94)(52)+3(32)(54)−558x^3 = \frac{27}{8} - 3\left(\frac{9}{4}\right)\left(\frac{\sqrt{5}}{2}\right) + 3\left(\frac{3}{2}\right)\left(\frac{5}{4}\right) - 5\frac{\sqrt{5}}{8}
    x3=278−27516+4516−558x^3 = \frac{27}{8} - \frac{27\sqrt{5}}{16} + \frac{45}{16} - \frac{5\sqrt{5}}{8}
    x3=27+4516−275+10516x^3 = \frac{27 + 45}{16} - \frac{27\sqrt{5} + 10\sqrt{5}}{16}
    x3=(3−52)3x^3 = \left(\frac{3 - \sqrt{5}}{2}\right)^300
    x3=(3−52)3x^3 = \left(\frac{3 - \sqrt{5}}{2}\right)^311
    Now we have the value of x3x^3 for the second solution.Now we need to find the value of x3=(3−52)3x^3 = \left(\frac{3 - \sqrt{5}}{2}\right)^333 for each solution of x3=(3−52)3x^3 = \left(\frac{3 - \sqrt{5}}{2}\right)^344.
    However, we realize that we made a mistake in the approach. We do not need to find the individual solutions for x3=(3−52)3x^3 = \left(\frac{3 - \sqrt{5}}{2}\right)^344 to calculate x3=(3−52)3x^3 = \left(\frac{3 - \sqrt{5}}{2}\right)^333 because we can use the given equation x3=(3−52)3x^3 = \left(\frac{3 - \sqrt{5}}{2}\right)^377 to simplify the expression directly.
    We need to correct our approach and use the given equation to simplify x3=(3−52)3x^3 = \left(\frac{3 - \sqrt{5}}{2}\right)^333 without finding the individual values of x3=(3−52)3x^3 = \left(\frac{3 - \sqrt{5}}{2}\right)^344.

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