If one of the root of the quadratic equation ax^2 + bx +c =0 is three times the other, show that 3b^2 = 16ac
ax^2 + bx + c = 0
x^2 + (b/a)x + (c/a) = 0
For quadratic equations with this form, the sum & product of roots are:
sum = b/a
product = c/a
Let R = one root
Let 3R = the other root (according to the problem)
Substituting to sum & product equations,
R + 3R = b/a
R(3R) = c/a
from the sum,
4R = b/a
R = b/(4a)
substituting this to the product,
R(3R) = c/a
3R^2 = c/a
3(b/(4a))^2 = c/a
3b^2 / (16a^2) = c/a
3b^2 = (16a^2)c/a
3b^2 = 16ac
Hope this helps~ :)
*sorry I mean sum = -b/a, and thus R = -b/(4a) . But the answer's still the same since it is squared.
To solve this problem, you need to apply the quadratic formula to the given equation and determine the roots.
The quadratic formula states that for any quadratic equation in the form ax^2 + bx + c = 0, the roots (or values of x) can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Let's assume that one of the roots is three times the other. This means that if x1 and x2 are the roots, we have the relationship:
x1 = 3x2
Substituting these values into the quadratic formula, we get:
(-b ± √(b^2 - 4ac)) / (2a) = 3(-b ± √(b^2 - 4ac)) / (2a)
To simplify the equation, we can cancel out the factors of 3:
-(-b ± √(b^2 - 4ac)) / (2a) = (-b ± √(b^2 - 4ac)) / (2a)
Now, let's focus on the numerator of the right-hand side of the equation:
-b ± √(b^2 - 4ac)
Expanding this expression, we have:
-b ± √(b^2 - 4ac) = ± √((b^2 - 4ac))
To remove the square root, we square both sides of the above equation:
(b ± √(b^2 - 4ac))^2 = (b^2 - 4ac)
Expanding the left-hand side, we get:
(b^2 ± 2b√(b^2 - 4ac) + (b^2 - 4ac) = b^2 - 4ac
Simplifying this equation, we find:
2b√(b^2 - 4ac) = -4ac
Dividing both sides by 2, we get:
b√(b^2 - 4ac) = -2ac
Squaring both sides once again, we have:
b^2(b^2 - 4ac) = 4a^2c
Simplifying the equation further:
b^4 - 4a^2c = 4a^2c
Rearranging the terms, we obtain:
b^4 = 16a^2c
Finally, we can express this equation in terms of 3b^2 and 16ac:
3b^2 = 16ac
Thus, we have shown that if one of the roots of the quadratic equation ax^2 + bx + c = 0 is three times the other, then 3b^2 = 16ac.