form the quadratic equations whose roots are(A:5 and -1/3
Answer my question
Sure, here's a quadratic equation with roots at 5 and -1/3:
(x - 5)(x + 1/3) = 0.
Now let's expand it and simplify:
x^2 + (1/3)x - 5x - (5/3) = 0.
Combining like terms:
x^2 - (14/3)x - (5/3) = 0.
Ta-da! That's your quadratic equation with roots at 5 and -1/3. But remember, I'm just a Clown Bot, so I hope you find this equation more comedic than mathematical!
To form a quadratic equation with roots A = 5 and A = -1/3, we can use the fact that the sum and product of the roots of a quadratic equation are related to the coefficients of the equation.
The sum of the roots is given by the equation: A + B = -b/a
where A and B are the roots and a and b are the coefficients of the quadratic equation Ax^2 + Bx + C = 0.
The product of the roots is given by the equation: A * B = c/a
where c is the constant term in the quadratic equation.
Let's substitute the given values in these equations:
For the sum of the roots:
5 + (-1/3) = -b/a
For the product of the roots:
5 * (-1/3) = c/a
Now, let's solve for the coefficients a, b, and c:
Using the sum of the roots equation, we have:
5 - 1/3 = -b/a
15/3 - 1/3 = -b/a
14/3 = -b/a
Multiplying through by 3 to eliminate the fraction:
14 = -3b/a
Simplifying, we have:
-3b/a = 14
Now, using the product of the roots equation, we have:
5 * (-1/3) = c/a
-5/3 = c/a
Multiplying through by 3 to eliminate the fraction:
-5 = 3c/a
Simplifying, we have:
3c/a = -5
From these equations, we can see that a = 3, b = -14, and c = -5.
Therefore, the quadratic equation with roots A = 5 and A = -1/3 can be written as:
3x^2 - 14x - 5 = 0.
To form a quadratic equation with given roots, we can use the fact that if a number 'a' is a root of a quadratic equation, then (x - a) is a factor of that equation.
Given roots are 5 and -1/3.
So, the quadratic equation can be formed using these roots as follows:
(x - 5)(x - (-1/3)) = 0
Removing the brackets:
(x - 5)(x + 1/3) = 0
Multiplying both terms by the common denominator (3) to eliminate the fractions:
3(x - 5)(3x + 1) = 0
Expanding:
3(x^2 - 5x + (1/3)x - 5(1/3)) = 0
3(x^2 - (14/3)x - (5/3)) = 0
Therefore, the quadratic equation with the given roots is:
3x^2 - (14/3)x - (5/3) = 0