Solve for x,leave your answer in surd form. 2/x+2 + 3/x+1=5
X=12/x-9 leave your answer in surds form
To solve the equation 2/(x+2) + 3/(x+1) = 5 for x, we need to find a common denominator and combine the fractions:
Multiplying the first fraction by (x+1) and the second fraction by (x+2), we get:
(2(x+1))/((x+2)(x+1)) + (3(x+2))/((x+1)(x+2)) = 5
Simplifying the denominators, the equation becomes:
(2x + 2)/(x^2 + 3x + 2) + (3x + 6)/(x^2 + 3x + 2) = 5
Combining the fractions, we have:
(2x + 2 + 3x + 6)/(x^2 + 3x + 2) = 5
Simplifying the numerator:
(5x + 8)/(x^2 + 3x + 2) = 5
Now we can cross multiply:
5(x^2 + 3x + 2) = (5x + 8)
Expanding the equation:
5x^2 + 15x + 10 = 5x + 8
Bringing all terms to one side:
5x^2 + 15x - 5x + 10 - 8 = 0
5x^2 + 10x + 2 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula:
Since the equation cannot be factored easily, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
For our equation, a = 5, b = 10, and c = 2:
x = (-10 ± √(10^2 - 4(5)(2)))/(2(5))
x = (-10 ± √(100 - 40))/10
x = (-10 ± √60)/10
x = (-10 ± 2√15)/10
Simplifying further:
x = -1 ± √15/5
Hence, the solution for x in surd form is:
x = (-1 + √15)/5 or x = (-1 - √15)/5
To solve for x in the equation (2/(x+2)) + (3/(x+1)) = 5, we need to clear the denominators and simplify the equation. Here's how to do it step by step:
Step 1: Clear the denominators by multiplying both sides of the equation by (x+2)(x+1). This will eliminate the fractions.
(x+2)(x+1) * (2/(x+2) + 3/(x+1)) = (x+2)(x+1) * 5
Simplifying the left side of the equation by distributing:
2(x+1) + 3(x+2) = 5(x+2)(x+1)
Step 2: Expanding the expression on the left side:
2x + 2 + 3x + 6 = 5(x^2 + 3x + 2x + 2)
Combine like terms:
5x + 8 = 5(x^2 + 5x + 2)
Step 3: Expanding the expression on the right side:
5x + 8 = 5x^2 + 25x + 10
Step 4: Rearrange the equation to form a quadratic equation in standard form (ax^2 + bx + c = 0):
0 = 5x^2 + 25x + 10 - 5x - 8
Combine like terms:
0 = 5x^2 + 20x + 2
Step 5: Set the equation equal to zero and simplify if necessary:
5x^2 + 20x + 2 = 0
Now, we have a quadratic equation that we can solve. However, it doesn't seem like the equation will factor nicely, so we will use the quadratic formula to find the roots.
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, the quadratic equation is 5x^2 + 20x + 2 = 0, so:
a = 5, b = 20, c = 2
Plugging these values into the quadratic formula:
x = (-20 ± √(20^2 - 4 * 5 * 2)) / (2 * 5)
Simplifying:
x = (-20 ± √(400 - 40)) / 10
x = (-20 ± √360) / 10
x = (-20 ± √(4 * 9 * 10)) / 10
x = (-20 ± 2√90) / 10
Therefore, the solution for x in surd form is:
x = (-20 + 2√90) / 10 or x = (-20 - 2√90) / 10