Question

2х³+mx²+13x+n=0m,n,x=? x1=2x2=3​

Answer 1

2х³+mx²+13x+n=0

m,n,x=?

x1=2

x2=3​

-------

x1 = 2

2*2³ + m*2² + 13*2 + n = 0

16 + 4m + 26 + n = 0

4m + n = - 42

x2 = 3

2*3³ + m*3² + 13*3 + n = 0

54 + 9m + 39 + n = 0

9m + n = -93

$\left \{ {{9m+n=-93} \atop {4m+n=-42}} \right.$

5m = -51

m = -10.2

4*(-10.2) + n = -42

n = -1.2

2(x - 2)(x - 3)(x - 0.1) = 0

x3 = 0.1

m,n,x= -10.2, -1.2, 0.1

Answer 2

To find the values of m, n, and x when you have the equations:

1. $2x^3 + mx^2 + 13x + n = 0$
2. $x_1 = 2$
3. $x_2 = 3$

First, let's use the given values of $x_1$ and $x_2$ to set up a system of equations:

1. When $x = x_1 = 2$: $2(2^3) + m(2^2) + 13(2) + n = 0$

2. When $x = x_2 = 3$: $2(3^3) + m(3^2) + 13(3) + n = 0$

Now, let's solve this system of equations to find the values of m and n:

1. $2(2^3) + m(2^2) + 13(2) + n = 16 + 4m + 26 + n = 0$

2. $2(3^3) + m(3^2) + 13(3) + n = 54 + 9m + 39 + n = 0$

Now, we have the following equations:

1. $4m + n = -42$
2. $9m + n = -93$

We can solve this system of equations simultaneously. Subtracting the first equation from the second equation eliminates n:

$(9m + n) - (4m + n) = -93 - (-42)$

Simplify:

$5m = -51$

Now, divide both sides by 5 to solve for m:

$m = -51/5$

Now that we have found the value of m, we can substitute it back into one of the equations to solve for n. Using the first equation:

$4m + n = -42$

$4(-51/5) + n = -42$

$-204/5 + n = -42$

Now, isolate n:

$n = -42 + 204/5$

To simplify the fraction on the right-hand side, find a common denominator:

$n = (-210/5) + (204/5)$

$n = -6/5$

So, the values of m and n are:

$m = -51/5$

$n = -6/5$

Now, we have the values of m and n. If you also need to find x, you can use either equation 1 or 2 to solve for x. Let's use equation 1:

$2x^3 + mx^2 + 13x + n = 0$

Plugging in the values of m and n:

$2x^3 - (51/5)x^2 + 13x - 6/5 = 0$

This is a cubic equation, and solving it for x might require numerical methods or a graphing calculator. You can approximate the solutions or use appropriate software to find the roots of the equation.