solve the following quadratic equation by completing the square
-x^(2)+3x-5=0
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:
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.
A student has asked the following question:
Solve the following quadratic equation by completing the square: -\(x^{2}\) + 3x - 5 = 0
To solve the quadratic equation by completing the square, we can follow these steps:
First, rearrange the equation into the standard form, making the coefficient of x2 positive:
\(x^{2} -3x+5=0\)
Next, move the constant term to the right side of the equation:
\(x^{2} -3x=-5\)
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:
\(x^{2} -3x+(\frac{3}{2})^{2} = -5+(\frac{3}{2})^{2}\)
Simplify the equation:
\(x^{2} -3x+\frac{9}{4}= -\frac{20}{4}+\frac{9}{4}\)
Combine the terms on the right side:
\(x^{2} -3x+\frac{9}{4}= -\frac{11}{4}\)
Now, write the left-hand side as a perfect square:
\((x-\frac{3}{2})^{2}= -\frac{11}{4}\)
Take the square root of both sides, being mindful of the plus-minus sign:
\(x-\frac{3}{2}= \pm\sqrt{-\frac{11}{4}}\)
\(x-\frac{3}{2}= \pm(i\frac{\sqrt{11}}{2} )\)
where \(i\) is the imaginary unit.
Finally, solve for x:
\(x=\frac{3}{2} \pm(\frac{\sqrt{11}}{2} )i\)
This shows the solutions in terms of complex numbers i.e. \(x=\frac{3}{2} + (\frac{\sqrt{11}}{2} )i \ \ \& \ \ x=\frac{3}{2} -(\frac{\sqrt{11}}{2} )i\)