If K = sqrt(1−3x) / −x, find the value(s) of x, such that K is: 1.1 Undefined, 1.2 Real, 1.3 Positive.

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Question 1 Given K = (√(1−3x))/(−x). 1.1 Undefined. For K to be undefined, either the denominator is zero or the expression under the square root is negative. Step 1: Set the denominator to zero. −x = 0; x = 0 Step 2: Set the expression under the square root to be negative. 1−3x < 0 1 < 3x 1/3 < x Step 3: Combine the conditions. K is undefined if x=0 or x > 1/3. x=0 or x > 1/3 1.2 Real. For K to be real, the expression under the square root must be non-negative, and the denominator must be non-zero. Step 1: Ensure the expression under the square root is non-negative. 1−3x ≥ 0 1 ≥ 3x x ≤ 1/3 Step 2: Ensure the denominator is non-zero. −x ≠ 0 x ≠ 0 Step 3: Combine the conditions. K is real if x ≤ 1/3 and x ≠ 0. x ≤ 1/3, x ≠ 0 1.3 Positive. For K to be positive, K > 0. This means (√(1−3x))/(−x) > 0. The numerator √(1−3x) must be real and positive, and the denominator −x must also be positive. Step 1: Ensure the numerator is positive. For √(1−3x) to be positive, 1−3x > 0. 1 > 3x x < 1/3 Step 2: Ensure the denominator is positive. −x > 0 x < 0 Step 3: Combine the conditions. Both conditions x < 1/3 and x < 0 must be met. The stricter condition is x < 0. x < 0 Question 2 2.1 Simplify (3x−2)(9x²+6x+4) Step 1: Recognize the pattern as the difference of cubes formula, (a−b)(a²+ab+b²) = a³−b³. Here, a=3x and b=2. (3x−2)((3x)² + (3x)(2) + 2²) Step 2: Apply the formula. (3x)³ − 2³ Step 3: Calculate the powers. 27x³ − 8 27x³ − 8 2.2 Simplify 2x⁻² ÷ x⁻⁵ + (x⁻¹)⁻³ Step 1: Simplify the division term using the exponent rule a^m ÷ aⁿ = a^(m−n). 2x⁻² ÷ x⁻⁵ = 2x⁻²⁻⁽⁻⁵⁾ = 2x⁻²⁺⁵ = 2x³; Step 2: Simplify the second term using the exponent rule (a^m)ⁿ = a^(mn). (x⁻¹)⁻³ = x^((−1) × (−3)) = x³ Step 3: Add the simplified terms. 2x³ + x³ = 3x³ 3x³ 2.3 Simplify (a⁻²−b⁻²) ÷ (a⁻¹+b⁻¹) Step 1: Rewrite terms with negative exponents as fractions. a⁻²−b⁻² = (1)/(a²) − (1)/(b²) = (b²−a²)/(a²b²); a⁻¹+b⁻¹ = 1/a + 1/b = (b+a)/(ab) Step 2: Perform the division by multiplying by the reciprocal of the divisor. (b²−a²)/(a²b²) ÷ (b+a)/(ab) = (b²−a²)/(a²b²) × (ab)/(b+a); Step 3: Factor the difference of squares in the numerator, b²−a² = (b−a)(b+a). ((b−a)(b+a))/(a²b²) × (ab)/(b+a) Step 4: Cancel common factors (b+a) and ab. ((b−a))/(ab) (b−a)/(ab) Question 3 3.1 Factorise the following expression fully: 5x³ + (40)/(x³) Step 1: Factor out the common factor of 5. 5(x³ + (8)/(x³)) Step 2: Recognize the expression inside the parenthesis as a sum of cubes, A³+B³ = (A+B)(A²−AB+B²), where A=x and B=2/x. 5(x³ + (2/x)³) Step 3: Apply the sum of cubes formula. 5(x + 2/x)(x² − x(2/x) + (2/x)²) Step 4: Simplify the terms inside the second parenthesis. 5(x + 2/x)(x² − 2 + (4)/(x²)) 5(x + 2/x)(x² − 2 + (4)/(x²)) 3.2 Solve simultaneously for x and y if: x²(y−3)+36(3−y)=0 Step 1: Rewrite the term 36(3−y) to have a common factor with x²(y−3). Note that (3−y) = −(y−3). x²(y−3) − 36(y−3) = 0 Step 2: Factor out the common term (y−3). (y−3)(x²−36) = 0 Step 3: Set each factor equal to zero to find the possible values for x and y. y−3 = 0 or x²−36 = 0 Step 4: Solve for y. y = 3 Step 5: Solve for x. x² = 36; x = ±√(36); x = ± 6; The solutions are y=3 (for any real x) or x=6 (for any real y) or x=−6 (for any real y). More precisely, the equation holds if y=3 (and x can be any real number) OR if x=6 or x=−6 (and y can be any real number). y=3 or x=6 or x=−6

If K = sqrt(1−3x) / -x, find the value(s) of x, such that K is: 1.1 Undefined, 1.2 Real, 1.3 Positive.

Answers

Step-by-step solution

Part 1

Part 1.1· 3 stepsx=0 or x>13x=0 \text{ or } x > \frac{1}{3}x=0 or x>31​

Set the denominator to zero.

Set the expression under the square root to be negative.

Combine the conditions.

Part 1.2· 3 stepsx≤13,x≠0x \le \frac{1}{3}, x \ne 0x≤31​,x=0

Ensure the expression under the square root is non-negative.

Ensure the denominator is non-zero.

Combine the conditions.

Part 1.3· 3 stepsx<0x < 0x<0

Ensure the numerator is positive.

Ensure the denominator is positive.

Combine the conditions.

Part 2

Part 2.1· 3 steps27x3−827x^3 - 827x3−8

Recognize the pattern as the difference of cubes formula, (a−b)(a2+ab+b2)=a3−b3(a-b)(a^2+ab+b^2) = a^3-b^3(a−b)(a2+ab+b2)=a3−b3.

Apply the formula.

Calculate the powers.

Part 2.2· 3 steps3x33x^33x3

Simplify the division term using the exponent rule am÷an=am−na^m \div a^n = a^{m-n}am÷an=am−n.

Simplify the second term using the exponent rule (am)n=amn(a^m)^n = a^{mn}(am)n=amn.

Add the simplified terms.

Part 2.3· 4 stepsb−aab\frac{b-a}{ab}abb−a​

Rewrite terms with negative exponents as fractions.

Perform the division by multiplying by the reciprocal of the divisor.

Factor the difference of squares in the numerator, b2−a2=(b−a)(b+a)b^2-a^2 = (b-a)(b+a)b2−a2=(b−a)(b+a).

Cancel common factors (b+a)(b+a)(b+a) and ababab.

Part 3

Part 3.1· 4 steps5(x+2x)(x2−2+4x2)5\left(x + \frac{2}{x}\right)\left(x^2 - 2 + \frac{4}{x^2}\right)5(x+x2​)(x2−2+x24​)

Factor out the common factor of 5.

Recognize the expression inside the parenthesis as a sum of cubes, A3+B3=(A+B)(A2−AB+B2)A^3+B^3 = (A+B)(A^2-AB+B^2)A3+B3=(A+B)(A2−AB+B2), where A=xA=xA=x and B=2xB=\frac{2}{x}B=x2​.

Apply the sum of cubes formula.

Simplify the terms inside the second parenthesis.

Part 3.2· 5 stepsy=3 or x=6 or x=−6y=3 \text{ or } x=6 \text{ or } x=-6y=3 or x=6 or x=−6

Rewrite the term 36(3−y)36(3-y)36(3−y) to have a common factor with x2(y−3)x^2(y-3)x2(y−3).

Factor out the common term (y−3)(y-3)(y−3).

Set each factor equal to zero to find the possible values for xxx and yyy.

Solve for yyy.

Solve for xxx.

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More Mathematics Questions

Part 1.1· 3 steps
$x=0 \text{ or } x > \frac{1}{3}$

Part 1.2· 3 steps
$x \le \frac{1}{3}, x \ne 0$

Part 1.3· 3 steps
$x < 0$

Part 2.1· 3 steps
$27x^3 - 8$

Recognize the pattern as the difference of cubes formula, $(a-b)(a^2+ab+b^2) = a^3-b^3$ .

Part 2.2· 3 steps
$3x^3$

Simplify the division term using the exponent rule $a^m \div a^n = a^{m-n}$ .

Simplify the second term using the exponent rule $(a^m)^n = a^{mn}$ .

Part 2.3· 4 steps
$\frac{b-a}{ab}$

Factor the difference of squares in the numerator, $b^2-a^2 = (b-a)(b+a)$ .

Cancel common factors $(b+a)$ and $ab$ .

Part 3.1· 4 steps
$5\left(x + \frac{2}{x}\right)\left(x^2 - 2 + \frac{4}{x^2}\right)$

Recognize the expression inside the parenthesis as a sum of cubes, $A^3+B^3 = (A+B)(A^2-AB+B^2)$ , where $A=x$ and $B=\frac{2}{x}$ .

Part 3.2· 5 steps
$y=3 \text{ or } x=6 \text{ or } x=-6$

Rewrite the term $36(3-y)$ to have a common factor with $x^2(y-3)$ .

Factor out the common term $(y-3)$ .

Set each factor equal to zero to find the possible values for $x$ and $y$ .

Solve for $y$ .

Solve for $x$ .