Find the polynomials roots to each of the following problems:
#1) x^2+3x+1
#2) x^2+4x+3=0
#3) -2x^2+4x-5
#3 is not an equation. Dod you omit "= 0" at the end?
#2 can be factored into (x+1)(x+3) = 0, so the roots are x=-1 and -3.
#1 Use the quadratic equation for the roots
[-b +/- sqrt (b^2-4ac)]/2a = [-3 +/-sqrt5]/2
"Roots" are solutions to an equation, or those particular values of the variable which make the equation a true statement.
Only your second problem is an equation, I will assume you meant the other two to be equations as well.
There are three common methods to solve quadratic equations:
1. by factoring. Only quadratic equations with rational roots will factor, so they are the exception
2. by completing the square. If the coefficient of the square term is 1, and the middle coefficient is even, this is a very good and fast method.
3. by the quadratic equation formuls. This is a sure method that works for all real and complex roots.
I will solve the #2 equation in those three ways.
x^2 + 4x + 3 = 0
(x+1)(x+3)=0
x+1=0 or x+3=0
so x= -1 or x=-3
x^2 + 4x + 3 = 0
x^2 + 4x + ..... = -3
x^2 + 4x + 4 = -3 + 4
(x + 2)^2 = 1
x+2 = ±1
x = -1 or x=-3
x^2 + 4x + 3 = 0
x = (-4 ±sqrt(16-4*1*3))/2
x = (-4 ± 2)/2
x=-1 or x=-3
For problem #1, the equation is x^2 + 3x + 1. To find the roots of this equation, we can use the quadratic equation formula:
x = [-b +/- sqrt(b^2 - 4ac)] / 2a
In this equation, a, b, and c are the coefficients of the quadratic equation. For the given equation, a = 1, b = 3, and c = 1. Plugging these values into the formula, we get:
x = [-3 +/- sqrt(3^2 - 4(1)(1))] / 2(1)
Simplifying further, we have:
x = [-3 +/- sqrt(9 - 4)] / 2
x = [-3 +/- sqrt(5)] / 2
So the roots of the equation x^2 + 3x + 1 are x = (-3 + sqrt(5)) / 2 and x = (-3 - sqrt(5)) / 2.
For problem #2, the equation is x^2 + 4x + 3 = 0. Using the factoring method, we can factor this equation as:
(x + 1)(x + 3) = 0
Setting each factor equal to 0, we get:
x + 1 = 0 or x + 3 = 0
Solving for x, we have:
x = -1 or x = -3
Therefore, the roots of the equation x^2 + 4x + 3 = 0 are x = -1 and x = -3.
As for problem #3, you mentioned that it is not an equation as it is missing the "= 0" at the end. Could you please provide the correct equation so that I can help you find its roots?