Question
Find the value of the constant K for which the equation 2x^2+(k+3)x+2x=0
Answers
GPT 3.5
To find the value of the constant K, we need to set the equation equal to zero and solve for K.
Given equation: 2x^2 + (k+3)x + 2x = 0
Simplify by combining like terms: 2x^2 + (k+5)x = 0
Now, we set the equation equal to zero: 2x^2 + (k+5)x = 0
To find the value of the constant K, we need to set the discriminant of the quadratic equation equal to zero, as the equation has no real roots.
The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by b^2 - 4ac.
In this case, a = 2, b = (k+5), and c = 0.
So, the discriminant is: (k+5)^2 - 4(2)(0)
Expand and simplify: k^2 + 10k + 25
Set the discriminant equal to zero: k^2 + 10k + 25 = 0
Factor the quadratic equation: (k+5)(k+5) = 0
(k+5)^2 = 0
Take the square root of both sides: k + 5 = 0
Subtract 5 from both sides: k = -5
Therefore, the constant K is -5 for the given equation.
Given equation: 2x^2 + (k+3)x + 2x = 0
Simplify by combining like terms: 2x^2 + (k+5)x = 0
Now, we set the equation equal to zero: 2x^2 + (k+5)x = 0
To find the value of the constant K, we need to set the discriminant of the quadratic equation equal to zero, as the equation has no real roots.
The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by b^2 - 4ac.
In this case, a = 2, b = (k+5), and c = 0.
So, the discriminant is: (k+5)^2 - 4(2)(0)
Expand and simplify: k^2 + 10k + 25
Set the discriminant equal to zero: k^2 + 10k + 25 = 0
Factor the quadratic equation: (k+5)(k+5) = 0
(k+5)^2 = 0
Take the square root of both sides: k + 5 = 0
Subtract 5 from both sides: k = -5
Therefore, the constant K is -5 for the given equation.