(please solve step by step)
solving equation (6x+2)/4 + (2x²-1)/(2x²+2) = (10x-1)/4x we got roots as
Well, let's start by simplifying the equation:
(6x + 2)/4 + (2x^2 - 1)/(2x^2 + 2) = (10x - 1)/4x
First, let's find a common denominator for the fractions in the equation. The common denominator will be 4(2x^2 + 2) for the first term and 4x(2x^2 + 2) for the second term. After we find the common denominators, we can add the fractions.
((6x + 2)(2x^2 + 2))/(4(2x^2 + 2)) + ((2x^2 - 1)(4x))/(4x(2x^2 + 2)) = (10x - 1)/4x
Simplifying further:
(12x^3 + 12x + 4x^2 + 4)/(8x^2 + 8) + (8x^3 - 4x)/(8x^3 + 8x) = (10x - 1)/4x
Now we can combine like terms and simplify the equation:
(12x^3 + 4x^2 + 12x + 4 + 8x^3 - 4x)/(8x^2 + 8) = (10x - 1)/4x
(20x^3 + 4x^2 + 8x)/(8x^2 + 8) = (10x - 1)/4x
Next, we can cross multiply to eliminate the denominators:
(20x^3 + 4x^2 + 8x)(4x) = (10x - 1)(8x^2 + 8)
80x^4 + 16x^3 + 32x^2 = 80x^3 - 8x^2 + 80x^2 - 8
Now, let's simplify and rearrange the equation:
80x^4 + 16x^3 + 32x^2 - 80x^3 + 8x^2 - 80x^2 + 8 = 0
80x^4 - 64x^3 - 40x^2 + 8 = 0
We have a quartic equation now, and solving it step by step can be quite complex. Unfortunately, I can't solve quartic equations in this format. However, there are several methods, such as factoring, synthetic division, or using numerical methods like Newton's method or the Ferrari method, that can be used to find the roots of this equation.
To solve the equation step by step:
Step 1: Simplify the equation by finding a common denominator for all the terms. The common denominator for the fractions is 4(2x² + 2), which simplifies to 8x² + 8.
(6x+2)/(4) + (2x²-1)/(2x²+2) = (10x-1)/(4x)
Step 2: Multiply the first fraction by (2x² + 2)/(2x² + 2) and the second fraction by 4/(4) to eliminate the denominators.
[(6x+2)(2x² + 2)]/(4)(2x² + 2) + [(2x²-1)(4)]/(4)(2x²+2) = [(10x-1)(4)]/(4x)
Simplifying further:
(6x² + 6x + 4x + 4)/(8x² + 8) + (8x² - 4)/(8x² + 8) = (40x - 4)/(4x)
Step 3: Combine like terms in both the numerator and denominator.
(6x² + 10x + 4)/(8x² + 8) + (8x² - 4)/(8x² + 8) = (40x - 4)/(4x)
Step 4: Combine the fractions on the left side of the equation by finding a common denominator.
[(6x² + 10x + 4) + (8x² - 4)]/(8x² + 8) = (40x - 4)/(4x)
Simplifying further:
(14x^2 + 10x)/(8x^2 + 8) = (40x - 4)/(4x)
Step 5: Multiply both sides of the equation by (8x^2 + 8) and (4x) to eliminate the denominators.
[(14x^2 + 10x)(4x)] = (40x - 4)(8x^2 + 8)
Simplifying further:
56x^3 + 40x^2 = 320x^3 - 32x^2 + 320x^2 - 32x
Step 6: Combine like terms on both sides of the equation.
56x^3 + 40x^2 = 320x^3 - 32x^2 + 320x^2 - 32x
56x^3 + 40x^2 = 320x^3 + 288x^2 - 32x
Step 7: Move all the terms to one side of the equation and arrange in decreasing order of power.
56x^3 + 40x^2 - 320x^3 - 288x^2 + 32x = 0
-264x^3 - 248x^2 + 32x = 0
Step 8: Factor out common factors.
8x(-33x^2 - 31x + 4) = 0
Step 9: Solve each factor separately.
First factor: 8x = 0
From this, we get x = 0.
Second factor: -33x^2 - 31x + 4 = 0
This quadratic equation can be factored or solved using the quadratic formula.
Calculating the discriminant: b^2 - 4ac
Discriminant = (-31)^2 - 4(-33)(4) = 961 + 528 = 1489
Since the discriminant is positive, the quadratic equation has two distinct real roots.
Using the quadratic formula: x = (-b ± √(b^2 - 4ac))/(2a)
x = (-(-31) ± √(961 - 4(-33)(4)))/(2(-33))
x = (31 ± √(961 + 528))/(2(-33))
x = (31 ± √1489)/(-66)
This gives us two more roots, which can be written as:
x = (31 + √1489)/(-66)
x = (31 - √1489)/(-66)
Therefore, the roots of the equation are:
x = 0, (31 + √1489)/(-66), and (31 - √1489)/(-66).
To solve the equation (6x+2)/4 + (2x²-1)/(2x²+2) = (10x-1)/4x, follow these steps:
Step 1: Simplify the equation by finding a common denominator for the fractions on both sides. The common denominator in this case is 4x(2x²+2). Multiply each fraction by the appropriate factors to get the common denominator:
(6x+2)*(2x²+2)/(4*(2x²+2)) + (2x²-1)*4x/(4x*(2x²+2)) = (10x-1)*4/(4x*(2x²+2))
Simplifying further:
(2(6x+2)x²+2(6x+2))/(4*(2x²+2)) + (8x²-4x)/(4x*(2x²+2)) = (40x-4)/(4x*(2x²+2))
Step 2: Expand and simplify the numerators:
(12x³+4x²+12x+4)/(4*(2x²+2)) + (8x²-4x)/(4x*(2x²+2)) = (40x-4)/(4x*(2x²+2))
Step 3: Combine the fractions on the left side by adding their numerators:
(12x³+4x²+12x+4+8x²-4x)/(4*(2x²+2)) = (40x-4)/(4x*(2x²+2))
Simplifying further:
(12x³+12x+12x²+4-4x)/(4*(2x²+2)) = (40x-4)/(4x*(2x²+2))
Step 4: Combine like terms in the numerator:
(12x³+12x²+8x)/(4*(2x²+2)) = (40x-4)/(4x*(2x²+2))
Step 5: Cancel out common factors:
(3x³+3x²+2x)/(2*(2x²+2)) = (10x-1)/(x*(2x²+2))
Step 6: Cross-multiply:
(3x³+3x²+2x)*x*(2x²+2) = (10x-1)*2*(2x²+2)
(3x^4+3x³+2x²)*(2x²+2) = (20x-2)*(2x²+2)
Step 7: Expand both sides:
6x^6+6x⁵+4x⁴+4x²+6x⁴+6x³+4x²+4x = 40x³-4x+40x²-4
6x^6+6x⁵+10x⁴+8x³+8x²+0x+0x = 40x³-4x+40x²-4
Step 8: Combine like terms:
6x^6+6x⁵+10x⁴+8x³+8x² = 40x³-4x+40x²-4
6x^6+6x⁵+10x⁴+8x³+8x²-40x³-4x+40x²-4 = 0
Step 9: Rearrange the terms and combine like terms:
6x^6+6x⁵-30x³+50x⁴+48x²-4x-4 = 0
The equation is now in standard form and can be further simplified or solved using algebraic methods, numerical methods, or technology. From here, you can try factoring, using the rational root theorem, or using a graphing calculator to find the roots.
(6x+2)/4 + (2x²-1)/(2x²+2) = (10x-1)/4x
Clear fractions by using the common denominator of 4x(x²+1). This requires that x≠0.
(6x+2)(x(x²+1)) + (2x²-1)(2x) = (10x-1)(x²+1)
(6x^4+2x^3+6x^2+2x) + (4x^3-2x) = (10x^3-x^2+10x-1)
6x^4 - 4x^3 + 7x^2 - 10x + 1 = 0 = 0
A little sniffing about with synthetic division shows that one root is x=1. So, now we have
(x-1)(6x^3+2x^2+9x-1) = 0
There are no other rational roots, so some kind of numeric method shows that another root is x = 0.1077
That leaves us with
6(x-1)(x-0.1077)(x^2+0.44x+1.55)
The quadratic has no real roots.