Solve using the Quadratic Formula (Find all complex-number solutions.) x2 + 4x + 6 = 0
I do not get this at all and this is just practice exercise. Please help as I have a test next week and if I do not get at least a C, I will not graduate. Thank you Thank you!!
For quadratic equation :
a x ^ 2 + b x + c = 0
The solutions are:
x1/2 = [ - b + OR - sqroot ( b ^ 2 - 4 a c ) ] / 2 a
Int this case :
a = 1
b = 4
c = 6
x1/2 = [ - 4 + OR - sqroot ( 4 ^ 2 - 4 * 1 * 6 ) ] / ( 2 * 1 )
x1/2 = [ - 4 + OR - sqroot ( 16 - 24 ) ] / 2
x1/2 = [ - 4 + OR - sqroot ( - 8 ) ] / 2
x1/2 = [ - 4 + OR - sqroot ( - 1 * 4 * 2 ) ] / 2
x1/2 = [ - 4 + OR - sqroot ( - 1 ) * sqroot ( 4 ) * sqroot ( 2 ) ] / 2
x1/2 = [ - 4 + OR - i * 2 * sqroot ( 2 ) ] / 2
x1/2 = [ - 4 + OR - 2 i * sqroot ( 2 ) ] / 2
x1/2 = - 4 / 2 + OR - [ 2 i * sqroot ( 2 ) ] / 2
x1/2 = - 2 + OR - i * sqroot ( 2 )
The solutions are:
x = - 2 - i * sqroot ( 2 )
and
x = - 2 + i * sqroot ( 2 )
Of course, I can help you solve this quadratic equation using the quadratic formula!
The quadratic formula is a formula that gives you the solutions of a quadratic equation in the form of ax^2 + bx + c = 0, where a, b, and c are coefficients. In this case, we have the equation x^2 + 4x + 6 = 0.
The quadratic formula states that the solutions (or roots) of the quadratic equation can be found using the following formula:
x = (-b ± √(b^2 - 4ac)) / (2a),
where ± represents taking both the positive and negative roots, and √(b^2 - 4ac) represents the square root of the discriminant.
Now let's apply the quadratic formula to solve the given equation:
a = 1, b = 4, c = 6.
Substituting these values into the formula:
x = (-4 ± √(4^2 - 4*1*6)) / (2*1),
Simplifying the expression further:
x = (-4 ± √(16 - 24)) / 2,
x = (-4 ± √(-8)) / 2.
Now we have a square root of a negative number, which means we will be dealing with complex-number solutions.
To simplify the square root of -8, we can rewrite it as the square root of -1 times the square root of 8. The square root of -1 is denoted by the imaginary unit "i". The square root of 8 can be written as 2√2.
Therefore, we have:
x = (-4 ± 2√2i) / 2,
Simplifying further:
x = -2 ± √2i.
This gives us the two complex-number solutions for the given equation as:
x = -2 + √2i, and
x = -2 - √2i.
And that's how you can use the quadratic formula to solve the quadratic equation x^2 + 4x + 6 = 0 and find the complex-number solutions. Make sure to double-check all the steps and calculations, especially when dealing with complex numbers. Good luck with your practice and upcoming test!