Answer :

To solve the quadratic equation -x²+3x-5=0 by completing the square, rearrange the equation, complete the square, factor the perfect square trinomial, and then solve for x.

Quadratic equation: -x²+3x-5=0

To solve by completing the square:

  1. Rearrange the equation to have x-terms on one side: x² - 3x = 5
  2. Complete the square by adding (3/2)² to both sides: x² - 3x + (3/2)² = 5 + (3/2)²
  3. Factor the perfect square trinomial: (x - 3/2)² = 5 + 9/4
  4. Solve for x
  5. final answer for this expression x = 3/2 ± √(5 + 9/4)

To solve the quadratic equation -x^2 + 3x - 5 = 0 by completing the square, first, rearrange it to make the coefficient of x^2 positive, then complete the square to derive the solutions in terms of complex numbers.

Explanation:

A student has asked the following question:

Solve the following quadratic equation by completing the square: -[tex]x^{2}[/tex] + 3x - 5 = 0

To solve the quadratic equation by completing the square, we can follow these steps:

  1. First, rearrange the equation into the standard form, making the coefficient of x2 positive:

    [tex]x^{2} -3x+5=0[/tex]

  2. Next, move the constant term to the right side of the equation:

    [tex]x^{2} -3x=-5[/tex]

  3. To complete the square, add and subtract (b/2)2 to the left side. Here, b is -3, so (b/2)2 = (3/2)2:

    [tex]x^{2} -3x+(\frac{3}{2})^{2} = -5+(\frac{3}{2})^{2}[/tex]

  4. Simplify the equation:

    [tex]x^{2} -3x+\frac{9}{4}= -\frac{20}{4}+\frac{9}{4}[/tex]

  5. Combine the terms on the right side:

    [tex]x^{2} -3x+\frac{9}{4}= -\frac{11}{4}[/tex]

  6. Now, write the left-hand side as a perfect square:

    [tex](x-\frac{3}{2})^{2}= -\frac{11}{4}[/tex]

  7. Take the square root of both sides, being mindful of the plus-minus sign:

    [tex]x-\frac{3}{2}= \pm\sqrt{-\frac{11}{4}}[/tex]

    [tex]x-\frac{3}{2}= \pm(i\frac{\sqrt{11}}{2} )[/tex]

    where [tex]i[/tex] is the imaginary unit.

  8. Finally, solve for x:

    [tex]x=\frac{3}{2} \pm(\frac{\sqrt{11}}{2} )i[/tex]

This shows the solutions in terms of complex numbers i.e. [tex]x=\frac{3}{2} + (\frac{\sqrt{11}}{2} )i \ \ \& \ \ x=\frac{3}{2} -(\frac{\sqrt{11}}{2} )i[/tex]

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