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4(1-t)=5t^(2) Let t=x and t=y be the solutions to the given equation. What is the value of -xy? ◻

4(1−t)=5t24(1-t)=5t^{2} \newlineLet t=xt=x and t=yt=y be the solutions to the given equation. \newlineWhat is the value of −xy-xy?\newline □\square

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Full solution

Q. 4(1−t)=5t24(1-t)=5t^{2} \newlineLet t=xt=x and t=yt=y be the solutions to the given equation. \newlineWhat is the value of −xy-xy?\newline □\square
  1. Rewrite in standard form: First, let's rewrite the equation in standard quadratic form.\newline4−4t=5t24 - 4t = 5t^2\newlineMove all terms to one side to set the equation to zero.\newline5t2+4t−4=05t^2 + 4t - 4 = 0
  2. Solve quadratic equation: Now, we need to solve the quadratic equation for tt. We can use the quadratic formula: t=−b±b2−4ac2at = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} Here, a=5a = 5, b=4b = 4, and c=−4c = -4.
  3. Plug values into formula: Plug the values into the quadratic formula.\newlinet = [−4±42−4⋅5⋅(−4)]/(2⋅5)-4 \pm \sqrt{4^2 - 4\cdot5\cdot(-4)}] / (2\cdot5)\newlinet = [−4±16+80]/10-4 \pm \sqrt{16 + 80}] / 10\newlinet = [−4±96]/10-4 \pm \sqrt{96}] / 10
  4. Find solutions for tt: Simplify the square root and the fraction.\newlinet=−4±4610t = \frac{-4 \pm 4\sqrt{6}}{10}\newlinet=−25±265t = -\frac{2}{5} \pm \frac{2\sqrt{6}}{5}\newlineSo, the solutions are t=−25+265t = -\frac{2}{5} + \frac{2\sqrt{6}}{5} and t=−25−265t = -\frac{2}{5} - \frac{2\sqrt{6}}{5}.
  5. Let xx and yy be: Let x=−25+265x = -\frac{2}{5} + \frac{2\sqrt{6}}{5} and y=−25−265y = -\frac{2}{5} - \frac{2\sqrt{6}}{5}. To find −xy-xy, we multiply xx and yy and then take the negative. -xy = -((-\frac{\(2\)}{\(5\)} + \frac{\(2\)\sqrt{\(6\)}}{\(5\)}) * (-\frac{\(2\)}{\(5\)} - \frac{\(2\)\sqrt{\(6\)}}{\(5\)}))
  6. Simplify multiplication: Use the difference of squares formula to simplify the multiplication.\(\newline−xy=−(((−25)2−(265)2))-xy = -(((-\frac{2}{5})^2 - (\frac{2\sqrt{6}}{5})^2))
  7. Calculate squares: Calculate the squares and simplify.\newline−xy=−(425−2425)-xy = -\left(\frac{4}{25} - \frac{24}{25}\right)\newline−xy=−(−2025)-xy = -\left(-\frac{20}{25}\right)
  8. Simplify negative sign: Simplify the negative sign and the fraction. \newline−xy=2025-xy = \frac{20}{25}\newline−xy=45-xy = \frac{4}{5}

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