solve: 5/2x+1/4x=9/4+x
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5/2x + 1/4x = 9/4 + x
Multiply both sides by 4x
10 + 1 = 9x + 4x^2
Bring all terms to one side of the equation.
4x^2 + 9x -11 = 0
Unfortunately, this equation does not factor. To get - 11, it factors into 11 and - 1 or -11 and 1. The other side (4x^2) can factor into both 2x or x and 4x. However, none of these combinations will give 9x. Do you have it written correctly?
Continue with abc formula so x = -9 + sqrt257 / 8 or x = -9 + sqrt257 / 8
To solve the equation 5/(2x) + 1/(4x) = 9/4 + x, you can follow these steps:
Step 1: Find a common denominator for the fractions on the left-hand side of the equation. In this case, the common denominator is 4x. Multiply the numerator and denominator of each fraction to obtain a common denominator:
(5 * 2) / (2x) + 1 / (4x) = 9/4 + x
10 / (4x) + 1 / (4x) = 9/4 + x
Step 2: Combine the fractions with the same denominator on the left-hand side of the equation:
(10 + 1) / (4x) = 9/4 + x
11 / (4x) = 9/4 + x
Step 3: Convert the mixed number on the right-hand side of the equation to an improper fraction. Multiply the whole number by the denominator and add the numerator, then place it over the denominator:
11 / (4x) = (9 + 4x) / 4
Step 4: Cross-multiply by multiplying the numerators of the fractions with the denominators of the opposite fractions:
11 * 4 = (9 + 4x) * (4x)
44 = (9 + 4x) * (4x)
Step 5: Expand the equation by multiplying each term within the parentheses by the factor outside:
44 = (9 * 4x) + (4x * 4x)
44 = 36x + 16x^2
Step 6: Rearrange the equation to form a quadratic equation in standard form (ax^2 + bx + c = 0):
16x^2 + 36x - 44 = 0
Step 7: Simplify the equation if possible, by dividing every term by their greatest common factor. In this case, it is 4:
4x^2 + 9x - 11 = 0
Now you have a quadratic equation. At this point, you can either factor the quadratic equation, complete the square, or use the quadratic formula to solve for x.
If you solve by factoring or the quadratic formula, you may find that x is a decimal or a fraction.