Form a quadratic equation in x whose roots are -3 and 1/4.
since you don't specify any particular form, just use the roots directly
(x+3)(4x-1) = 0
You can massage that as you will.
For quadratic equation:
a x² + b x + c = a ( x - x1 ) ( x - x2 )
where x1 and x2 are roots
In this case x1 = - 3 , y = 1 / 4
a x² + b x + c = a [ x - ( - 3 ) ] ( x - 1 / 4 ) =
a ( x + 3 ) ( x - 1 / 4 ) = a ( x² + 3 x - 1 / 4 x - 3 / 4 ) =
a ( x² + 12 / 4 x - 1 / 4 x - 3 / 4 ) = a ( x² + 11 / 4 x - 3 / 4 )
where a is some real number ( leading coefficient of quadratic equation )
To form a quadratic equation in x with roots -3 and 1/4, we can use the fact that the roots of a quadratic equation are the values of x that make the equation equal to zero.
The equation will take the form:
(x - root1)(x - root2) = 0
Substituting the values of the roots into the equation, we have:
(x - (-3))(x - 1/4) = 0
Simplifying, we get:
(x + 3)(x - 1/4) = 0
Expanding the equation, we have:
x(x) + x(-1/4) + 3(x) + 3(-1/4) = 0
x^2 - (1/4)x + 3x - 3/4 = 0
Combining like terms, we get:
x^2 + (11/4)x - 3/4 = 0
Therefore, the quadratic equation with roots -3 and 1/4 is:
x^2 + (11/4)x - 3/4 = 0
To form a quadratic equation when given the roots, you can use the fact that if r and s are the roots of a quadratic equation, then the equation can be written in the form:
(x - r)(x - s) = 0
In this case, the roots are -3 and 1/4.
So, the equation with these roots can be written as:
(x - (-3))(x - (1/4)) = 0
Simplifying this equation, we have:
(x + 3)(x - 1/4) = 0
Expanding the equation, we get:
x(x) + x(-1/4) + 3(x) + 3(-1/4) = 0
Simplifying further:
x^2 - (1/4)x + 3x - (3/4) = 0
Combining like terms:
x^2 + (11/4)x - (3/4) = 0
So, the quadratic equation with roots -3 and 1/4 is:
x^2 + (11/4)x - (3/4) = 0