one of the roots of the equation x^2 - 2qx + 2q +4 = 0 is three times the other root, determine the value q
one root
(2q+sqrt(4q^2-8q-16))/2
other root
(2q -sqrt(4q^2-8q-16))/2
but one root=3*other root or
(2q+sqrt(4q^2-8q-16)=6q-3sqrt(4q^2-8q-16)
8q=-4sqrt( )
divide by 4, square both sides..
4q^2=4q^2-8q-16)
q=(-2)
check my work. Typing is hard to do without making a mistake.
recall that for ax^2 + bx + c = 0 ,
the sum of the roots = -b/a
product of the roots = c/a
we are given that the roots are r and 3r
so from your equation of
x^2 - 2qx + 2q+4 = 0
a = 1, b = -2q, c = 2q+4
r+3r = 2q/1
4r = 2q
q = 2r
r(3r) = 2q+4
3r^2 = 2(2r) + 4
3r^2 - 4r - 4 = 0
(r-2)(3r +2) = 0
r = 2 or r = -2/3
if r = 2, q = 4
ir r = -2/3) , q = -4/3
check for q=4, then the equation is
x^2 - 8x + 12 = 0
(x-2)(x-6) = 0
x = 2 or x = 6
and one root is 3 times the other.
checking for q = -4/3 is a bit more messy, but still varifies.
To determine the value of q, we need to use the information given about the roots of the equation. Let's proceed step by step:
Step 1: Identify the roots of the equation.
Let the roots of the equation be denoted as r1 and r2.
Step 2: Translate the given information into equations.
According to the problem, one of the roots (let's say r1) is three times the other root (r2). This can be expressed as:
r1 = 3r2
Step 3: Solve the equation.
We have a quadratic equation of the form ax^2 + bx + c = 0.
Comparing the given equation x^2 - 2qx + 2q + 4 = 0 with the standard form, we can identify that:
a = 1, b = -2q, c = 2q + 4
Using the formula for the sum and product of roots, we can write:
r1 + r2 = -b/a
r1 * r2 = c/a
Substituting the values from our equation, we have:
3r2 + r2 = -(-2q)/1 (r1 + r2 = -b/a)
(3r2) * r2 = (2q + 4)/1 (r1 * r2 = c/a)
Simplifying these equations, we get:
4r2 = 2q
3r2^2 = 2q + 4
Step 4: Solve the system of equations.
Substitute the value of 2q from the first equation into the second equation to eliminate q:
3r2^2 = 4r2 + 4
Rearrange the equation to form a quadratic equation:
3r2^2 - 4r2 - 4 = 0
Step 5: Solve the quadratic equation.
This quadratic equation can be solved using factoring, completing the square, or the quadratic formula. We will use the quadratic formula:
r2 = (-b ± √(b^2 - 4ac))/2a
For our equation, a = 3, b = -4, and c = -4, so:
r2 = (-(-4) ± √((-4)^2 - 4*3*(-4)))/(2*3)
r2 = (4 ± √(16 + 48))/6
r2 = (4 ± √64)/6
r2 = (4 ± 8)/6
This gives two possible values for r2:
r2 = (4 + 8)/6 = 2
r2 = (4 - 8)/6 = -2/3
Step 6: Calculate the value of q.
Using the equation 4r2 = 2q, substitute in the value of r2 to find q:
4(2) = 2q -> 8 = 2q
q = 8/2
q = 4
Therefore, the value of q is 4.