Alright, let's solve the given equation step-by-step.
We start with the equation:
[tex]\[ x^2 + \frac{33}{8} x = \frac{9}{8} x \][/tex]
First, let's simplify the equation by moving all terms to one side:
[tex]\[ x^2 + \frac{33}{8}x - \frac{9}{8}x = 0 \][/tex]
Combine the like terms:
[tex]\[ x^2 + \left(\frac{33}{8} - \frac{9}{8}\right)x = 0 \][/tex]
Simplify the coefficient of [tex]\( x \)[/tex]:
[tex]\[ x^2 + \frac{24}{8}x = 0 \][/tex]
[tex]\[ x^2 + 3x = 0 \][/tex]
Next, we factor the equation:
[tex]\[ x(x + 3) = 0 \][/tex]
Now, we set each factor equal to zero to find the solutions:
1. [tex]\( x = 0 \)[/tex]
2. [tex]\( x + 3 = 0 \Rightarrow x = -3 \)[/tex]
Thus, the solutions to the equation are:
[tex]\[ x = 0, -3 \][/tex]
Let's verify these solutions by substituting them back into the original equation.
1. Substitute [tex]\( x = 0 \)[/tex] into the original equation:
[tex]\[ 0^2 + \frac{33}{8}(0) = \frac{9}{8}(0) \][/tex]
[tex]\[ 0 = 0 \][/tex]
This is true.
2. Substitute [tex]\( x = -3 \)[/tex] into the original equation:
[tex]\[ (-3)^2 + \frac{33}{8}(-3) = \frac{9}{8}(-3) \][/tex]
[tex]\[ 9 - \frac{99}{8} = -\frac{27}{8} \][/tex]
Convert 9 to a fraction with denominator 8:
[tex]\[ \frac{72}{8} - \frac{99}{8} = -\frac{27}{8} \][/tex]
[tex]\[ \frac{72 - 99}{8} = -\frac{27}{8} \][/tex]
This is true.
Therefore, the roots of the equation [tex]\( x^2 + \frac{33}{8} x = \frac{9}{8} x \)[/tex] are:
[tex]\[ x = 0, -3 \][/tex]
