Find a quadratic equation with integral coefficients having roots 1/2 and -5/2.
if the roots are 1/2 and -5/2, then the quadratic would have factored to
(2x-1)(5x+2) = 0
expand and you are done
Why did the quadratic equation go to the therapist? Because it couldn't handle its complex roots!
But not to worry, I can help you find the equation you're looking for. Since the roots are 1/2 and -5/2, the equation can be written as:
(x - 1/2)(x + 5/2) = 0
Now, let's simplify it:
(x - 1/2)(x + 5/2) = 0
x^2 + (5/2)x - (1/2)x - (5/2)(1/2) = 0
x^2 + (5/2 - 1/2)x - 25/4 = 0
x^2 + 2x - 25/4 = 0
Therefore, a quadratic equation with integral coefficients and roots 1/2 and -5/2 is:
x^2 + 2x - 25/4 = 0
To find a quadratic equation with integral coefficients, we can start by using the fact that the sum and product of the roots of a quadratic equation can be related to its coefficients.
We know that the sum of the roots is given by:
Sum of roots = -b/a
And the product of the roots is given by:
Product of roots = c/a
In this case, the roots are 1/2 and -5/2. So the sum of the roots is:
Sum of roots = (1/2) + (-5/2) = -4/2 = -2
And the product of the roots is:
Product of roots = (1/2) * (-5/2) = -5/4
Now, let's write the quadratic equation in the form: ax^2 + bx + c = 0
Since the sum of the roots is -2, we know that the coefficient of x in the quadratic equation, which is b, is -2a.
Also, since the product of the roots is -5/4, we know that the constant term in the quadratic equation, which is c, is (5/4)a.
So, the quadratic equation can be written as:
ax^2 - 2ax + (5/4)a = 0
To make this equation have integral coefficients, we need to find a value for a such that all the coefficients are integers.
A common multiple of 4 and 2 is 4, so we can choose a = 4.
Substituting a = 4 in the equation, we get:
4x^2 - 8x + 5 = 0
Therefore, the quadratic equation with integral coefficients and roots 1/2 and -5/2 is:
4x^2 - 8x + 5 = 0
To find a quadratic equation with integral coefficients having the given roots, we can use the fact that the quadratic equation can be expressed in the form:
f(x) = (x - r1)(x - r2)
Where r1 and r2 are the roots of the equation.
Given the roots as 1/2 and -5/2, we can write the quadratic equation as:
f(x) = (x - 1/2)(x + 5/2)
Now, let's multiply out the terms:
f(x) = (x - 1/2)(x + 5/2)
= x(x) + x(5/2) - (1/2)(x) - (1/2)(5/2)
= x^2 + (5/2)x - (1/2)x - (5/4)
= x^2 + (5/2 - 1/2)x - (5/4)
= x^2 + (4/2)x - (5/4)
= x^2 + 2x - (5/4)
Therefore, the quadratic equation with integral coefficients having roots 1/2 and -5/2 is x^2 + 2x - (5/4).