Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Ask your questions and receive accurate answers from professionals with extensive experience in various fields on our platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
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]
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]
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.