solve the equation x³ − 7x² + 36 = 0 , given that one root is double the other
3,6,-2
To solve the equation x³ - 7x² + 36 = 0, given that one root is double the other, we can use the information that one root is double the other to set up a system of equations.
Let's assume that the roots of the equation are a and 2a, where a is a constant.
Using Vieta's formulas, the sum of the roots of a cubic equation is equal to the negative coefficient of the x² term divided by the coefficient of the x³ term. In this case, the sum of the roots is:
a + 2a = -(-7)/1
3a = 7
a = 7/3
Now that we know the value of a, we can find the roots of the equation.
The equation is x³ - 7x² + 36 = 0.
Substituting a = 7/3, we get:
(7/3)³ - 7(7/3)² + 36 = 0
Simplifying this equation, we have:
(343/27) - 343/9 + 36 = 0
Multiplying everything by 27 to eliminate the denominators, we get:
343 - 1029 + 972 = 0
Reordering the terms:
343 + 972 - 1029 = 0
Simplifying further:
1314 - 1029 = 0
285 = 0
This statement is false. Therefore, there is no solution to the equation x³ - 7x² + 36 = 0 where one root is double the other.
To solve the equation x³ − 7x² + 36 = 0, given that one root is double the other, we can start by assuming that the two roots are a and 2a, where a is a real number.
With this assumption, we can rewrite the equation as follows:
(x - a)(x - 2a)(x - 2a) = 0
Expanding this equation gives:
(x - a)(x² - 4ax + 4a²) = 0
Now, we can distribute (x - a) to both terms inside the parentheses:
x³ - 4ax² + 4a³ - ax² + 4a²x - 4a³ = 0
Combine like terms:
x³ - (4a + a)x² + (4a² - 4a³) = 0
Simplify further:
x³ - 5ax² + 4a² - 4a³ = 0
Comparing this equation with the given equation x³ − 7x² + 36 = 0, we can see that:
-5a = -7 (coefficient of x²)
4a² - 4a³ = 36 (constant term)
Solve the first equation for a:
-5a = -7
a = -7 / -5
a = 7/5 or 1.4
Now substitute the value of a back into the second equation to find the constant term:
4(7/5)² - 4(7/5)³ = 36
4(49/25) - 4(343/125) = 36
(196/25) - (1372/125) = 36
[(196*5) - (1372*25)] / 125 = 36
(980 - 34300) / 125 = 36
(980 - 34300) = 36*125
-33320 = 36*125
-33320 = 4500
Since -33320 is not equal to 4500, we have encountered a contradiction. Therefore, there is no real solution to the equation x³ - 7x² + 36 = 0 with one root being double the other.
if the roots are a,2a,b then
a+2a+b = 7
2a^2b = -36
a*2a + ab + 2ab = 0
or, a(2a+3b) = 0
so, b = -2a/3
Solve those and you find 3,6,-2