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Solve. 10=-4x+3x^(2) Choose 1 answer: (A) x=1,-(1)/(2) (B) x=(-1+-sqrt10)/(2) (C) x=(-2+-sqrt34)/(-3) (D) x=(-3+-sqrt17)/(2)

Solve.\newline10=−4x+3x2 10=-4 x+3 x^{2} \newlineChoose 11 answer:\newline(A) x=1,−12 x=1,-\frac{1}{2} \newline(B) x=−1±102 x=\frac{-1 \pm \sqrt{10}}{2} \newline(C) x=−2±34−3 x=\frac{-2 \pm \sqrt{34}}{-3} \newline(D) x=−3±172 x=\frac{-3 \pm \sqrt{17}}{2}

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Q. Solve.\newline10=−4x+3x2 10=-4 x+3 x^{2} \newlineChoose 11 answer:\newline(A) x=1,−12 x=1,-\frac{1}{2} \newline(B) x=−1±102 x=\frac{-1 \pm \sqrt{10}}{2} \newline(C) x=−2±34−3 x=\frac{-2 \pm \sqrt{34}}{-3} \newline(D) x=−3±172 x=\frac{-3 \pm \sqrt{17}}{2}
  1. Rewrite equation in standard form: Rewrite the equation in standard quadratic form, ax2+bx+c=0ax^2 + bx + c = 0.\newlineWe have 10=−4x+3x210 = -4x + 3x^2. To rewrite it, we move all terms to one side to get 0=3x2−4x−100 = 3x^2 - 4x - 10.
  2. Identify coefficients: Identify the coefficients aa, bb, and cc from the standard form.\newlineComparing 0=3x2−4x−100 = 3x^2 - 4x - 10 with ax2+bx+c=0ax^2 + bx + c = 0, we get:\newlinea=3a = 3\newlineb=−4b = -4\newlinec=−10c = -10
  3. Substitute values into quadratic formula: Substitute the values of aa, bb, and cc into the quadratic formula x=−b±b2−4ac2ax = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}.\newlineSubstitute a=3a = 3, b=−4b = -4, and c=−10c = -10 into the formula.\newlinex=−(−4)±(−4)2−4⋅3⋅(−10)2⋅3x = \frac{{-(-4) \pm \sqrt{{(-4)^2 - 4 \cdot 3 \cdot (-10)}}}}{{2 \cdot 3}}
  4. Simplify expression under square root: Simplify the expression under the square root and the constants outside the square root.\newlinex=4±16+1206x = \frac{{4 \pm \sqrt{{16 + 120}}}}{6}\newlinex=4±1366x = \frac{{4 \pm \sqrt{{136}}}}{6}\newlinex=4±16⋅8.56x = \frac{{4 \pm \sqrt{{16 \cdot 8.5}}}}{6}\newlinex=4±48.56x = \frac{{4 \pm 4\sqrt{{8.5}}}}{6}
  5. Factor out common factor in numerator: Simplify the expression further by factoring out the common factor in the numerator.\newlinex=4(1±8.5)6x = \frac{4(1 \pm \sqrt{8.5})}{6}\newlineSince 44 and 66 have a common factor of 22, we can simplify the fraction by dividing both the numerator and the denominator by 22.\newlinex=2(1±8.5)3x = \frac{2(1 \pm \sqrt{8.5})}{3}
  6. Two possible solutions for x: Now we have two possible solutions for x.\newlinex = (2+28.53)(\frac{2 + 2\sqrt{8.5}}{3}) or x = (2−28.53)(\frac{2 - 2\sqrt{8.5}}{3})\newlineThese solutions can be further simplified to:\newlinex = (23)+28.53(\frac{2}{3}) + \frac{2\sqrt{8.5}}{3} or x = (23)−28.53(\frac{2}{3}) - \frac{2\sqrt{8.5}}{3}
  7. Compare solutions with answer choices: Compare the solutions with the given answer choices.\newlineThe solutions we have are in the form x=23±28.53x = \frac{2}{3} \pm \frac{2\sqrt{8.5}}{3}, which is not immediately recognizable in the answer choices. However, we can simplify 8.5\sqrt{8.5} to 172\sqrt{\frac{17}{2}} to see if it matches any of the choices.\newlinex=23±21723x = \frac{2}{3} \pm \frac{2\sqrt{\frac{17}{2}}}{3}\newlinex=23±343x = \frac{2}{3} \pm \frac{\sqrt{34}}{3}\newlineThis matches answer choice (C) x=−2±34−3x = \frac{-2 \pm \sqrt{34}}{-3}.

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