To write a quadratic equation given the roots, we can use the fact that if a quadratic equation has roots [tex]\( r_1 \)[/tex] and [tex]\( r_2 \)[/tex], then the equation can be written in the form:
[tex]\[ a(x - r_1)(x - r_2) = 0 \][/tex]
where [tex]\( a \)[/tex] is the leading coefficient.
Given the roots 3 and 1, and the leading coefficient 4, we can substitute these values into the formula:
[tex]\[ 4(x - 3)(x - 1) = 0 \][/tex]
Now, we need to expand this to get the quadratic equation in the standard form ([tex]\( ax^2 + bx + c = 0 \)[/tex]).
First, expand the factors within the parentheses:
[tex]\[ (x - 3)(x - 1) = x^2 - x - 3x + 3 \][/tex]
Simplify and combine like terms:
[tex]\[ x^2 - 4x + 3 \][/tex]
Now, multiply this by the leading coefficient, which is 4:
[tex]\[ 4(x^2 - 4x + 3) = 4x^2 - 16x + 12 \][/tex]
So the quadratic equation in the standard form, with a leading coefficient of 4 and roots of 3 and 1, is:
[tex]\[ 4x^2 - 16x + 12 = 0 \][/tex]
This is the answer to the question, a quadratic equation with the required properties.
