1)3x^2-12x-21 solve by completing the square in simplest radical form
2) (1/x+3)-(2/3-x)=(4/x^2-9)find values of x
3)3/radical(3a^2b^4) is equivalent to
4) factor completely 27x^-75x^4
1) To solve the quadratic equation 3x^2 - 12x - 21 by completing the square in simplest radical form, follow these steps:
Step 1: Move the constant term to the right side of the equation:
3x^2 - 12x = 21
Step 2: Divide the entire equation by the coefficient of x^2, which is 3, to simplify the equation:
x^2 - 4x = 7
Step 3: Take half of the coefficient of x (in this case, -4) and square it:
-4/2 = -2
(-2)^2 = 4
Step 4: Add the square from Step 3 to both sides of the equation:
x^2 - 4x + 4 = 7 + 4
x^2 - 4x + 4 = 11
Step 5: Rewrite the left side of the equation as a perfect square:
(x - 2)^2 = 11
Step 6: Take the square root of both sides of the equation:
√(x - 2)^2 = √11
Step 7: Solve for x:
x - 2 = ±√11
Step 8: Add 2 to both sides of the equation:
x = 2 ± √11
Therefore, the solution to the equation is x = 2 ± √11.
2) To find the values of x in the equation (1/x + 3) - (2/3 - x) = (4/x^2 - 9), follow these steps:
Step 1: Simplify both sides of the equation by finding a common denominator:
Multiply all terms by x(3 - x) to get rid of the denominators.
x(3 - x)(1/x + 3) - x(3 - x)(2/3 - x) = x(3 - x)(4/x^2 - 9)
Step 2: Distribute and simplify each term:
(3 - x) + x(2) = 4(3 - x)(x) - 9(3 - x)(x^2)
Step 3: Expand and simplify each term:
3 - x + 2x = 12x - 4x^2 - 27x^3 + 9x^2
Step 4: Combine like terms:
3 + x = -15x^3 + 10x^2
Step 5: Rearrange the equation to set it equal to zero:
-15x^3 + 10x^2 - x + 3 = 0
Step 6: Solve the equation using factoring, the rational root theorem, or a calculator to find the possible rational roots. Once you have found a root, you can use synthetic division to simplify the equation.
Step 7: Repeat Step 6 until you have found all the roots.
Step 8: Once you have found the values of x, substitute them back into the original equation to check if they satisfy the equation.
3) To simplify 3/(√(3a^2b^4)), follow these steps:
Step 1: Simplify the expression under the radical:
√(3a^2b^4) = √(3 * a^2 * b^2 * b^2) = ab^2√3
Step 2: Substitute the simplified expression back into the original expression:
3/(√(3a^2b^4)) = 3/(ab^2√3)
Therefore, the simplified expression is 3/(ab^2√3).
4) To factor completely the expression 27x^-75x^4, follow these steps:
Step 1: Rewrite the expression with a common factor:
27x^-75x^4 = 27x^4 (x^-75)
Step 2: Use the power rule for exponents to simplify the expression:
27x^4 (x^-75) = 27(x^4/x^75) = 27/x^71
Therefore, the factored form of 27x^-75x^4 is 27/x^71.