form a quadratic equation whose roots are 2.5+root3 and 2.5-root3 giving your answer in the form ax^2+bx+c=0 where a, b and c are integers.
y = (x - (5/2 + √3))(x - (5/2 - √3))
y = ((x - 5/2)^2 - 3)
y = x^2 - 5x + 13/4
y = 4x^2 - 20x + 13
To form a quadratic equation with the given roots, we can use the fact that if a quadratic equation has roots α and β, then the equation can be written as:
(x - α)(x - β) = 0
In this case, the roots are 2.5 + √3 and 2.5 - √3. So, we have:
(x - (2.5 + √3))(x - (2.5 - √3)) = 0
Expanding this equation gives us:
(x - 2.5 - √3)(x - 2.5 + √3) = 0
Now, we can simplify the equation further using the difference of two squares:
[(x - 2.5) - √3][(x - 2.5) + √3] = 0
Using the difference of two squares formula (a^2 - b^2 = (a + b)(a - b)), we can rewrite the equation as:
[(x - 2.5)^2 - (√3)^2] = 0
(x - 2.5)^2 - 3 = 0
Expanding the square gives:
x^2 - 5x + 6.25 - 3 = 0
Finally, combining like terms, we get the quadratic equation in the required form:
x^2 - 5x + 3.25 = 0
Multiplying the equation by 4 to get rid of decimals, we multiply all terms by 4:
4x^2 - 20x + 13 = 0
Therefore, the quadratic equation with roots 2.5 + √3 and 2.5 - √3 is:
4x^2 - 20x + 13 = 0