Solve for x: 4x⁴ - 12x³ = 0 Using Common Factor Method

Question

4x412x3=0 4x^4-12x^3=0

Solve the equation above for x.

Video Solution

Solution Steps

00:00 Find X
00:03 Factor with the term 4X cubed
00:11 Take out the common factor from the parentheses
00:21 We want to find which solution zeroes each factor in the product
00:25 This is one solution
00:31 Now let's find the second solution
00:37 And this is the solution to the question

Step-by-Step Solution

Shown below is the given problem:

4x412x3=0 4x^4-12x^3=0

First, note that on the left side we are able factor the expression by using a common factor. The largest common factor for the numbers and variables in this case is 4x3 4x^3 due to the fact that the third power is the lowest power in the equation. Therefore it is included in both the term with the fourth power and the term with the third power. Any power higher than this is not included in the term with the third power, which is the lowest. Hence this is the term with the highest power that can be factored out as a common factor for the variables,

For the numbers, note that 12 is a multiple of 4, therefore 4 is the largest common factor for the numbers in both terms of the expression,

Let's continue to factor the expression:

4x412x3=04x3(x3)=0 4x^4-12x^3=0 \\ \downarrow\\ 4x^3(x-3)=0

Proceed to the left side of the equation that we obtained in the last step. There is a multiplication of algebraic expressions and on the right side the number 0. Therefore given that the only way to obtain 0 from a multiplication is to multiply by 0, at least one of the expressions in the multiplication on the left side must equal zero,

Meaning:

4x3=0/:4x3=0/3x=0 4x^3=0 \hspace{8pt}\text{/}:4\\ x^3=0 \hspace{8pt}\text{/}\sqrt[3]{\hspace{6pt}}\\ \boxed{x=0}

In solving the equation above, we first divided both sides of the equation by the term with the unknown and then extracted a cube root for both sides of the equation.

(In this case, extracting an odd root for the right side of the equation yielded one possibility)

Or:

x3=0x=3 x-3=0\\ \boxed{x=3}

Let's summarize the solution of the equation:

4x412x3=04x3(x3)=04x3=0x=0x3=0x=3x=0,3 4x^4-12x^3=0 \\ \downarrow\\ 4x^3(x-3)=0 \\ \downarrow\\ 4x^3=0 \rightarrow\boxed{ x=0}\\ x-3=0\rightarrow \boxed{x=3}\\ \downarrow\\ \boxed{x=0,3}

Therefore the correct answer is answer A.

Answer

x=0,3 x=0,3