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Solve the equation by factoring: 34x^(2)-140 x-2x^(3)=0 Answer: x=

Solve the equation by factoring:\newline34x2−140x−2x3=0 34 x^{2}-140 x-2 x^{3}=0 \newlineAnswer: x= x=

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Q. Solve the equation by factoring:\newline34x2−140x−2x3=0 34 x^{2}-140 x-2 x^{3}=0 \newlineAnswer: x= x=
  1. Rewrite Equation in Standard Form: Rewrite the equation in standard form.\newlineTo solve the equation by factoring, we first need to rewrite the equation in standard form, which means the terms should be ordered from the highest power to the lowest power of xx.\newlineThe given equation is 34x2−140x−2x3=034x^2 - 140x - 2x^3 = 0. We need to rearrange the terms to get the cubic term first.\newlineThe standard form of the equation is −2x3+34x2−140x=0-2x^3 + 34x^2 - 140x = 0.
  2. Factor Out Common Factor: Factor out the greatest common factor.\newlineWe can see that each term in the equation has a factor of xx. We will factor out the greatest common factor, which is xx.\newlineThe factored equation is x(−2x2+34x−140)=0x(-2x^2 + 34x - 140) = 0.
  3. Factor Quadratic Expression: Factor the quadratic expression.\newlineNow we need to factor the quadratic expression −2x2+34x−140-2x^2 + 34x - 140. To do this, we look for two numbers that multiply to −2×−140=280-2 \times -140 = 280 and add up to 3434.\newlineThe numbers that satisfy these conditions are 4040 and −7-7.\newlineWe can rewrite the quadratic expression as −2x2+40x−7x−140-2x^2 + 40x - 7x - 140.
  4. Group and Factor by Grouping: Group the terms and factor by grouping.\newlineWe group the terms to factor by grouping: (−2x2+40x)+(−7x−140)(-2x^2 + 40x) + (-7x - 140).\newlineNow we factor out the common factors from each group: 2x(−x+20)−7(−x−20)2x(-x + 20) - 7(-x - 20).
  5. Factor Out Common Binomial Factor: Factor out the common binomial factor.\newlineWe notice that both groups have a common binomial factor of (−x+20)(-x + 20).\newlineThe fully factored form of the quadratic expression is (2x−7)(−x+20)(2x - 7)(-x + 20).
  6. Write Fully Factored Form: Write the fully factored form of the original equation.\newlineNow we combine the factored quadratic expression with the xx we factored out in Step 22.\newlineThe fully factored form of the original equation is x(2x−7)(−x+20)=0x(2x - 7)(-x + 20) = 0.
  7. Solve for x: Solve for x by setting each factor equal to zero.\newlineTo find the solutions, we set each factor equal to zero and solve for xx.\newlinex=0x = 0, 2x−7=02x - 7 = 0, and −x+20=0-x + 20 = 0.\newlineSolving each equation gives us the solutions: x=0x = 0, x=72x = \frac{7}{2}, and x=20x = 20.

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