Sure, let's solve the equation step-by-step:
Given the equation:
[tex]\[ x\left(\frac{360}{x} - \frac{48}{60}\right) + 5\left(\frac{360}{x} - \frac{48}{60}\right) = 360 \][/tex]
First, simplify the fractions inside the parentheses:
[tex]\[ \frac{48}{60} = 0.8 \][/tex]
Therefore, the equation becomes:
[tex]\[ x\left(\frac{360}{x} - 0.8\right) + 5\left(\frac{360}{x} - 0.8\right) = 360 \][/tex]
Next, distribute [tex]\( x \)[/tex] and [tex]\( 5 \)[/tex] across the terms in the parentheses:
[tex]\[ x \cdot \frac{360}{x} - x \cdot 0.8 + 5 \cdot \frac{360}{x} - 5 \cdot 0.8 = 360 \][/tex]
Simplify each term:
[tex]\[ 360 - 0.8x + \frac{1800}{x} - 4 = 360 \][/tex]
Combine like terms on the left-hand side:
[tex]\[ 360 + \frac{1800}{x} - 0.8x - 4 = 360 \][/tex]
Simplify further:
[tex]\[ 356 + \frac{1800}{x} - 0.8x = 360 \][/tex]
Next, move 356 to the right-hand side by subtracting 356 from both sides:
[tex]\[ \frac{1800}{x} - 0.8x = 4 \][/tex]
Now multiply every term by [tex]\( x \)[/tex] to clear the fraction (assuming [tex]\( x \neq 0 \)[/tex]):
[tex]\[ 1800 - 0.8x^2 = 4x \][/tex]
Rearrange the terms to form a standard quadratic equation:
[tex]\[ 0.8x^2 + 4x - 1800 = 0 \][/tex]
To make the equation simpler, multiply through by 10 to clear the decimal:
[tex]\[ 8x^2 + 40x - 18000 = 0 \][/tex]
This is a quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex]. Let's use the quadratic formula to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\( a = 8 \)[/tex], [tex]\( b = 40 \)[/tex], and [tex]\( c = -18000 \)[/tex]. Plugging these values into the formula:
[tex]\[ x = \frac{-40 \pm \sqrt{40^2 - 4(8)(-18000)}}{2(8)} \][/tex]
[tex]\[ x = \frac{-40 \pm \sqrt{1600 + 576000}}{16} \][/tex]
[tex]\[ x = \frac{-40 \pm \sqrt{577600}}{16} \][/tex]
[tex]\[ x = \frac{-40 \pm 760}{16} \][/tex]
This gives us two solutions:
[tex]\[ x = \frac{720}{16} = 45 \][/tex]
[tex]\[ x = \frac{-800}{16} = -50 \][/tex]
So, the solutions to the equation are:
[tex]\[ x = 45 \quad \text{and} \quad x = -50 \][/tex]
