One solution of
12x^3-23x^2-3x+2=0 is 2. Find all other solutions.
I tried rearranging the equation to factor by grouping, but that 23x^2 is bothersome and I can't get the right answers (I know the answers are 2, 1/4, and -1/3).
Help?
Thank you!
since you were told that x=2 is a solution, you know that x-2 must be a factor and must divide evenly into your expression.
So use either long division or synthetic division to change your equation to
(x-2)(12x^2 ...........) = 0
since your second factor is a quadratic you can use the quadratic formula to solve that.
(But since you know the solutions, we can conclude that the other factors have to be (4x-1)(3x+1) )
To find all the solutions of the given equation 12x^3 - 23x^2 - 3x + 2 = 0, you can use a combination of factoring and synthetic division. Let's go step by step:
Step 1: Check if you can factor out the common factor
In some cases, you might be able to factor out a common factor from all the terms in an equation. However, in this case, there is no common factor in the equation that can be factored out.
Step 2: Use Rational Root Theorem to find possible roots
The Rational Root Theorem states that if a rational number p/q (where p is a factor of the constant term and q is a factor of the leading coefficient) is a root of a polynomial equation, then p is a factor of the constant term, and q is a factor of the leading coefficient.
In this case, the constant term is 2 (which can only be factored into 1 and 2), and the leading coefficient is 12 (which can be factored into 1, 2, 3, 4, 6, and 12). Therefore, the possible rational roots are ±1, ±2, or ±1/2, ±1/3, ±2/3, ±1/4, ±1/6, ±1/12.
Step 3: Try possible roots using synthetic division
To find the remaining roots, we can use synthetic division to test each possible root (starting with the easiest one, which is 2 in this case). Assuming 2 is a root, we divide the polynomial equation by (x-2).
2 | 12 -23 -3 2
24 2 -2
_____________________
12 1 -1 (Remainder)
The result after synthetic division is 12x^2 + x - 1. Now, the equation can be rewritten as (x-2)(12x^2 + x - 1) = 0.
Step 4: Solve the remaining quadratic equation
To find the remaining roots, we need to solve the quadratic equation 12x^2 + x - 1 = 0. There are several ways to solve it, such as factoring, completing the square, or using the quadratic formula.
In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
where a = 12, b = 1, and c = -1.
Plugging in the values, we have:
x = (-1 ± √(1^2 - 4(12)(-1))) / (2(12))
x = (-1 ± √(1 + 48)) / 24
x = (-1 ± √49) / 24
Simplifying further, we get:
x = (-1 ± 7) / 24
So the remaining roots are:
x = (-1 + 7) / 24 = 6/24 = 1/4
x = (-1 - 7) / 24 = -8/24 = -1/3
Therefore, the solutions of the given equation are x = 2, 1/4, and -1/3.
To find the other solutions of the equation 12x^3-23x^2-3x+2=0, we can use the fact that if a value, say "a", is a solution to a polynomial equation, then (x - a) is a factor of the polynomial.
Since 2 is given as a solution, we know that (x - 2) is a factor of the polynomial. So, let's perform polynomial division using long division or synthetic division to find the other factor and ultimately, the other solutions.
Using synthetic division:
```
2 | 12 -23 -3 2
| 24 2 -2
_____________________________
12 1 -1 0
```
So, the quotient is 12x^2 + x - 1. Now, let's solve this quadratic equation to find the remaining solutions.
To solve 12x^2 + x - 1 = 0, we can use factoring or the quadratic formula. Since factoring might not be easy in this case, let's use the quadratic formula.
The quadratic formula is given by:
```
x = (-b ± √(b^2 - 4ac)) / 2a
```
For our quadratic equation, a = 12, b = 1, and c = -1.
```
x = (-1 ± √(1^2 - 4(12)(-1))) / 2(12)
```
Simplifying further:
```
x = (-1 ± √(1 + 48)) / 24
x = (-1 ± √49) / 24
x = (-1 ± 7) / 24
```
Therefore, the other two solutions are:
```
x = (-1 + 7) / 24 = 6 / 24 = 1 / 4
x = (-1 - 7) / 24 = -8 / 24 = -1 / 3
```
So, the solutions to the equation 12x^3-23x^2-3x+2=0 are 2, 1/4, and -1/3.