Solve x² + 144 = 24x: Complete Quadratic Equation Guide

Question

Solve the following problem:

x2+144=24x x^2+144=24x

Video Solution

Solution Steps

00:00 Find X
00:03 Arrange the equation so that the right side equals 0
00:11 Factor X squared
00:16 Factor 144 into 12 squared
00:21 Factor 24X into factors 2, 12, and X
00:25 Use the shortened multiplication formulas and find the parentheses
00:33 Isolate X
00:38 And this is the solution to the question

Step-by-Step Solution

Proceed to solve the given equation:

x2+144=24x x^2+144=24x

Arrange the equation by moving terms:

x2+144=24xx224x+144=0 x^2+144=24x \\ x^2-24x+144=0

Note that we are able to factor the expression on the left side by using the perfect square trinomial formula:

(ab)2=a22ab+b2 (\textcolor{red}{a}-\textcolor{blue}{b})^2=\textcolor{red}{a}^2-2\textcolor{red}{a}\textcolor{blue}{b}+\textcolor{blue}{b}^2

As demonstrated below:

144=122 144=12^2

Therefore, we'll represent the rightmost term as a squared term:

x224x+144=0x224x+122=0 x^2-24x+144=0 \\ \downarrow\\ \textcolor{red}{x}^2-24x+\textcolor{blue}{12}^2=0

Now let's examine once again the perfect square trinomial formula mentioned earlier:

(ab)2=a22ab+b2 (\textcolor{red}{a}-\textcolor{blue}{b})^2=\textcolor{red}{a}^2-\underline{2\textcolor{red}{a}\textcolor{blue}{b}}+\textcolor{blue}{b}^2

And the expression on the left side in the equation that we obtained in the last step:

x224x+122=0 \textcolor{red}{x}^2-\underline{24x}+\textcolor{blue}{12}^2=0

Note that the terms x2,122 \textcolor{red}{x}^2,\hspace{6pt}\textcolor{blue}{12}^2 indeed match the form of the first and third terms in the perfect square trinomial formula (which are highlighted in red and blue),

However, in order to factor this expression (which is on the left side of the equation) using the perfect square trinomial formula mentioned, the remaining term must also match the formula, meaning the middle term in the expression (underlined):

(ab)2=a22ab+b2 (\textcolor{red}{a}-\textcolor{blue}{b})^2=\textcolor{red}{a}^2-\underline{2\textcolor{red}{a}\textcolor{blue}{b}}+\textcolor{blue}{b}^2

In other words - we'll query whether we can represent the expression on the left side of the equation as:

x224x+122=0?x22x12+122=0 \textcolor{red}{x}^2-\underline{24x}+\textcolor{blue}{12}^2=0\\ \updownarrow\text{?}\\ \textcolor{red}{x}^2-\underline{2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{12}}+\textcolor{blue}{12}^2=0

And indeed it is true that:

2x12=24x 2\cdot x\cdot12=24x

Therefore we can represent the expression on the left side of the equation as a perfect square trinomial:

x22x12+122=0(x12)2=0 \textcolor{red}{x}^2-\underline{2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{12}}+\textcolor{blue}{12}^2=0 \\ \downarrow\\ (\textcolor{red}{x}-\textcolor{blue}{12})^2=0

From here we can take the square root of both sides of the equation (and don't forget that there are two possibilities - positive and negative when taking an even root of both sides of an equation), then we'll easily solve by isolating the variable:

(x12)2=0/x12=±0x12=0x=12 (x-12)^2=0\hspace{8pt}\text{/}\sqrt{\hspace{6pt}}\\ x-12=\pm0\\ x-12=0\\ \boxed{x=12}

Let's summarize the solution of the equation:

x2+144=24xx224x+144=0x22x12+122=0(x12)2=0x12=0x=12 x^2+144=24x \\ x^2-24x+144=0 \\ \downarrow\\ \textcolor{red}{x}^2-2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{12}+\textcolor{blue}{12}^2=0 \\ \downarrow\\ (\textcolor{red}{x}-\textcolor{blue}{12})^2=0 \\ \downarrow\\ x-12=0\\ \downarrow\\ \boxed{x=12}

Therefore the correct answer is answer C.

Answer

x=12 x=12