Solve by Factoring:
1.) x²- x + 12 = 0
2.) 4x³ + 21x² - 18x = 0
3.) x³- x² - x + 1
1.
Factor by completing the square or the quadratic formula.
However, factoring by trial and error is faster, noting that only a few rational factors are possible, namely x±1, x±2, x±3, x±4, x±6.
2. note that x is a common factor of all the terms, so the problem reduces to a quadratic expression.
3. The sum of the coeffients is zero implies that (x-1) is a factor.
Reduce the expression to a quadratic by long division.
factoring usually assumes we are dealing with rational numbers
x^2 - x + 12 = 0 does not factor (there are no two numbers which multiply to get 12 and add to get -1)
2. x(4x^2 + 21x - 18)
= x(4x - 3)(x + 6)
3. grouping
x^2(x-1) - 1(x-1)
= (x-1)(x^1 - 1(
= (x-1)(x-1)(x+1) or (x+1)(x-1)^2
Thank you Reiny, I overlooked #1.
1x1=1x10000+9+9+9=10027
To solve these equations by factoring, follow these steps:
1.) x² - x + 12 = 0:
We look for two numbers whose product is equal to 12 and whose sum is equal to -1. The numbers are -3 and -4. So, we rewrite the equation:
(x - 4)(x - 3) = 0
By applying the zero-product property, we set each factor equal to zero:
x - 4 = 0 or x - 3 = 0
Solving each equation, we find:
x = 4 or x = 3
Therefore, the solutions are x = 4 and x = 3.
2.) 4x³ + 21x² - 18x = 0:
First, we factor out the greatest common factor, which is 4x:
4x(x² + (21/4)x - (18/4)) = 0
Simplifying, we have:
4x(x² + (21/4)x - 9/2) = 0
Next, we look for two numbers whose product is equal to -9/2 and whose sum is equal to 21/4. The numbers are 3/2 and -3. So, we rewrite the equation:
4x(x - 3)(x + (3/2)) = 0
By applying the zero-product property, we set each factor equal to zero:
4x = 0 or x - 3 = 0 or x + (3/2) = 0
Solving each equation, we find:
x = 0 or x = 3 or x = -3/2
Therefore, the solutions are x = 0, x = 3, and x = -3/2.
3.) x³ - x² - x + 1:
Unfortunately, this equation does not readily factor using integer coefficients. In cases like this, we can either use synthetic division or use numerical methods (such as the Rational Root Theorem or a graphing calculator) to approximate the solutions.