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Solve. -6x^(2)+6-2x=x Choose 1 answer: (A) x=(5+-sqrt57)/(16) (B) x=(-4+-sqrt34)/(3) (C) x=(-7+-3sqrt41)/(-16) (D) x=(1+-sqrt17)/(-4)

Solve.\newline−6x2+6−2x=x -6 x^{2}+6-2 x=x \newlineChoose 11 answer:\newline(A) x=5±5716 x=\frac{5 \pm \sqrt{57}}{16} \newline(B) x=−4±343 x=\frac{-4 \pm \sqrt{34}}{3} \newline(C) x=−7±341−16 x=\frac{-7 \pm 3 \sqrt{41}}{-16} \newline(D) x=1±17−4 x=\frac{1 \pm \sqrt{17}}{-4}

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Q. Solve.\newline−6x2+6−2x=x -6 x^{2}+6-2 x=x \newlineChoose 11 answer:\newline(A) x=5±5716 x=\frac{5 \pm \sqrt{57}}{16} \newline(B) x=−4±343 x=\frac{-4 \pm \sqrt{34}}{3} \newline(C) x=−7±341−16 x=\frac{-7 \pm 3 \sqrt{41}}{-16} \newline(D) x=1±17−4 x=\frac{1 \pm \sqrt{17}}{-4}
  1. Identify Quadratic Equation: Now we have a quadratic equation in the form ax2+bx+c=0ax^2 + bx + c = 0, where a=−6a = -6, b=−3b = -3, and c=6c = 6. We can solve this using the quadratic formula, x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
  2. Calculate Discriminant: Let's calculate the discriminant b2−4acb^2 - 4ac first.\newlineDiscriminant = (−3)2−4(−6)(6)(-3)^2 - 4(-6)(6)\newlineDiscriminant = 9−(−144)9 - (-144)\newlineDiscriminant = 9+1449 + 144\newlineDiscriminant = 153153
  3. Apply Quadratic Formula: Now we can plug the values of aa, bb, and the discriminant into the quadratic formula.\newlinex=−(−3)±1532×−6x = \frac{-(-3) \pm \sqrt{153}}{2 \times -6}\newlinex=3±153−12x = \frac{3 \pm \sqrt{153}}{-12}
  4. Simplify Square Root: Simplify the square root of the discriminant if possible. Since 153153 is not a perfect square, we leave it as 153\sqrt{153}.\newlinex=3±153−12x = \frac{3 \pm \sqrt{153}}{-12}
  5. Simplify Fraction: Now we need to simplify the fraction by dividing the numerator by the denominator.\newlinex=3−12±153−12x = \frac{3}{-12} \pm \frac{\sqrt{153}}{-12}\newlinex=−14±153−12x = \frac{-1}{4} \pm \frac{\sqrt{153}}{-12}
  6. Simplify Second Term: We can simplify the second term by dividing 153\sqrt{153} by −12-12.
    x=(−14)±(153−12)x = (-\frac{1}{4}) \pm (\frac{\sqrt{153}}{-12})
    x=(−14)±(−15312)x = (-\frac{1}{4}) \pm (-\frac{\sqrt{153}}{12})
  7. Write Possible Solutions: We can now write the two possible solutions for xx.x=(−14)+(−153)/12x = \left(-\frac{1}{4}\right) + \left(-\sqrt{153}\right)/12 or x=(−14)−(−153)/12x = \left(-\frac{1}{4}\right) - \left(-\sqrt{153}\right)/12
  8. Find Common Denominator: To match the answer choices, we need to find a common denominator for the two terms in each solution.\newlinex=(−3/12)−(153/12)x = (-3/12) - (\sqrt{153}/12) or x=(−3/12)+(153/12)x = (-3/12) + (\sqrt{153}/12)\newlinex=(−3−153)/12x = (-3 - \sqrt{153})/12 or x=(−3+153)/12x = (-3 + \sqrt{153})/12
  9. Match Answer Choices: Now we can see that the solutions match one of the answer choices. \newlinex=(−3−153)/12x = (-3 - \sqrt{153})/12 or x=(−3+153)/12x = (-3 + \sqrt{153})/12\newlineThis corresponds to answer choice (D) x=(1±17)/(−4)x = (1 \pm \sqrt{17})/(-4) if we simplify 153\sqrt{153} to 17∗9\sqrt{17*9} and take out a factor of 33 from the square root. However, this simplification is incorrect because 153\sqrt{153} cannot be simplified to 17∗9\sqrt{17*9} as 153153 is not a multiple of 99. Therefore, we have made a math error.

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