Solve the equation −x2−3x=5−3x
and re-express the answer as a complex number with the imaginary unit.
explain how you get the answer
To solve the equation -x^2 - 3x = 5 - 3x, we can start by subtracting 5 and adding 3x to both sides of the equation:
-x^2 - 3x - 3x = 5 - 3x - 3x
-x^2 - 3x - 3x + 6x = 5 - 6x
-x^2 = 5 - 6x
Next, we can move the -6x term to the left side by adding 6x to both sides:
-x^2 + 6x = 5
Now, we can rearrange the equation to have the quadratic term positive:
x^2 - 6x = -5
In this form, we can use the quadratic formula to solve for x. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 1, b = -6, and c = -5. Plugging these values into the quadratic formula:
x = (-(-6) ± √((-6)^2 - 4(1)(-5))) / (2(1))
x = (6 ± √(36 + 20)) / 2
x = (6 ± √56) / 2
Simplifying further:
x = (6 ± 2√14) / 2
x = 3 ± √14
So the solutions to the equation -x^2 - 3x = 5 - 3x are x = 3 + √14 and x = 3 - √14.
Now, to re-express these solutions as complex numbers with the imaginary unit, we can rewrite them as:
x = 3 + √14 + 0i
x = 3 - √14 + 0i
There is no imaginary part because the solutions do not involve the imaginary unit, i.