Answer :
Sure, let's solve the given equations step-by-step for the variable [tex]\( x \)[/tex]:
1. [tex]\(2.1 \quad x^2 = 100\)[/tex]
To isolate [tex]\( x^2 \)[/tex], we divide both sides of the equation by 2.1:
[tex]\[ x^2 = \frac{100}{2.1} \][/tex]
Solving for [tex]\( x \)[/tex], we take the square root of both sides:
[tex]\[ x = \sqrt{\frac{100}{2.1}} \quad \text{or} \quad x = -\sqrt{\frac{100}{2.1}} \][/tex]
Therefore, the solutions are:
[tex]\[ x = 6.900655593423542 \quad \text{or} \quad x = -6.900655593423542 \][/tex]
2. [tex]\(2.24 \times x^3 = 32\)[/tex]
We begin by isolating [tex]\( x^3 \)[/tex] on one side of the equation:
[tex]\[ x^3 = \frac{32}{2.24} \][/tex]
Next, we solve for [tex]\( x \)[/tex] by taking the cube root of both sides:
[tex]\[ x = \left( \frac{32}{2.24} \right)^{\frac{1}{3}} \][/tex]
Therefore, the solution is:
[tex]\[ x = 2.4264275032025866 \][/tex]
3. [tex]\(2.33 \times x^2 + 1 = 49\)[/tex]
We subtract 1 from both sides to isolate the [tex]\( x^2 \)[/tex] term:
[tex]\[ 2.33 \times x^2 = 48 \][/tex]
Next, divide both sides by 2.33:
[tex]\[ x^2 = \frac{48}{2.33} \][/tex]
Solving for [tex]\( x \)[/tex], we take the square root of both sides:
[tex]\[ x = \sqrt{\frac{48}{2.33}} \quad \text{or} \quad x = -\sqrt{\frac{48}{2.33}} \][/tex]
Therefore, the solutions are:
[tex]\[ x = 4.538816846833403 \quad \text{or} \quad x = -4.538816846833403 \][/tex]
4. [tex]\(2.4 \times x^2 + 1 = 5\)[/tex]
Subtract 1 from both sides to isolate the [tex]\( x^2 \)[/tex] term:
[tex]\[ 2.4 \times x^2 = 4 \][/tex]
Next, divide both sides by 2.4:
[tex]\[ x^2 = \frac{4}{2.4} \][/tex]
Solving for [tex]\( x \)[/tex], we take the square root of both sides:
[tex]\[ x = \sqrt{\frac{4}{2.4}} \quad \text{or} \quad x = -\sqrt{\frac{4}{2.4}} \][/tex]
Therefore, the solutions are:
[tex]\[ x = 1.2909944487358056 \quad \text{or} \quad x = -1.2909944487358056 \][/tex]
5. [tex]\(2.5 \times x^3 + 7 = 34\)[/tex]
Subtract 7 from both sides to isolate the [tex]\( x^3 \)[/tex] term:
[tex]\[ 2.5 \times x^3 = 27 \][/tex]
Next, divide both sides by 2.5:
[tex]\[ x^3 = \frac{27}{2.5} \][/tex]
Solving for [tex]\( x \)[/tex], we take the cube root of both sides:
[tex]\[ x = \left( \frac{27}{2.5} \right)^{\frac{1}{3}} \][/tex]
Therefore, the solution is:
[tex]\[ x = 2.2104188991842317 \][/tex]
6. [tex]\(2.64 \times x^3 - 3 = 29\)[/tex]
Add 3 to both sides to isolate the [tex]\( x^3 \)[/tex] term:
[tex]\[ 2.64 \times x^3 = 32 \][/tex]
Next, divide both sides by 2.64:
[tex]\[ x^3 = \frac{32}{2.64} \][/tex]
Solving for [tex]\( x \)[/tex], we take the cube root of both sides:
[tex]\[ x = \left( \frac{32}{2.64} \right)^{\frac{1}{3}} \][/tex]
Therefore, the solution is:
[tex]\[ x = 2.297111188608965 \][/tex]
In summary, the solutions for [tex]\( x \)[/tex] are:
1. [tex]\( x = 6.900655593423542 \quad \text{or} \quad x = -6.900655593423542 \)[/tex]
2. [tex]\( x = 2.4264275032025866 \)[/tex]
3. [tex]\( x = 4.538816846833403 \quad \text{or} \quad x = -4.538816846833403 \)[/tex]
4. [tex]\( x = 1.2909944487358056 \quad \text{or} \quad x = -1.2909944487358056 \)[/tex]
5. [tex]\( x = 2.2104188991842317 \)[/tex]
6. [tex]\( x = 2.297111188608965 \)[/tex]
1. [tex]\(2.1 \quad x^2 = 100\)[/tex]
To isolate [tex]\( x^2 \)[/tex], we divide both sides of the equation by 2.1:
[tex]\[ x^2 = \frac{100}{2.1} \][/tex]
Solving for [tex]\( x \)[/tex], we take the square root of both sides:
[tex]\[ x = \sqrt{\frac{100}{2.1}} \quad \text{or} \quad x = -\sqrt{\frac{100}{2.1}} \][/tex]
Therefore, the solutions are:
[tex]\[ x = 6.900655593423542 \quad \text{or} \quad x = -6.900655593423542 \][/tex]
2. [tex]\(2.24 \times x^3 = 32\)[/tex]
We begin by isolating [tex]\( x^3 \)[/tex] on one side of the equation:
[tex]\[ x^3 = \frac{32}{2.24} \][/tex]
Next, we solve for [tex]\( x \)[/tex] by taking the cube root of both sides:
[tex]\[ x = \left( \frac{32}{2.24} \right)^{\frac{1}{3}} \][/tex]
Therefore, the solution is:
[tex]\[ x = 2.4264275032025866 \][/tex]
3. [tex]\(2.33 \times x^2 + 1 = 49\)[/tex]
We subtract 1 from both sides to isolate the [tex]\( x^2 \)[/tex] term:
[tex]\[ 2.33 \times x^2 = 48 \][/tex]
Next, divide both sides by 2.33:
[tex]\[ x^2 = \frac{48}{2.33} \][/tex]
Solving for [tex]\( x \)[/tex], we take the square root of both sides:
[tex]\[ x = \sqrt{\frac{48}{2.33}} \quad \text{or} \quad x = -\sqrt{\frac{48}{2.33}} \][/tex]
Therefore, the solutions are:
[tex]\[ x = 4.538816846833403 \quad \text{or} \quad x = -4.538816846833403 \][/tex]
4. [tex]\(2.4 \times x^2 + 1 = 5\)[/tex]
Subtract 1 from both sides to isolate the [tex]\( x^2 \)[/tex] term:
[tex]\[ 2.4 \times x^2 = 4 \][/tex]
Next, divide both sides by 2.4:
[tex]\[ x^2 = \frac{4}{2.4} \][/tex]
Solving for [tex]\( x \)[/tex], we take the square root of both sides:
[tex]\[ x = \sqrt{\frac{4}{2.4}} \quad \text{or} \quad x = -\sqrt{\frac{4}{2.4}} \][/tex]
Therefore, the solutions are:
[tex]\[ x = 1.2909944487358056 \quad \text{or} \quad x = -1.2909944487358056 \][/tex]
5. [tex]\(2.5 \times x^3 + 7 = 34\)[/tex]
Subtract 7 from both sides to isolate the [tex]\( x^3 \)[/tex] term:
[tex]\[ 2.5 \times x^3 = 27 \][/tex]
Next, divide both sides by 2.5:
[tex]\[ x^3 = \frac{27}{2.5} \][/tex]
Solving for [tex]\( x \)[/tex], we take the cube root of both sides:
[tex]\[ x = \left( \frac{27}{2.5} \right)^{\frac{1}{3}} \][/tex]
Therefore, the solution is:
[tex]\[ x = 2.2104188991842317 \][/tex]
6. [tex]\(2.64 \times x^3 - 3 = 29\)[/tex]
Add 3 to both sides to isolate the [tex]\( x^3 \)[/tex] term:
[tex]\[ 2.64 \times x^3 = 32 \][/tex]
Next, divide both sides by 2.64:
[tex]\[ x^3 = \frac{32}{2.64} \][/tex]
Solving for [tex]\( x \)[/tex], we take the cube root of both sides:
[tex]\[ x = \left( \frac{32}{2.64} \right)^{\frac{1}{3}} \][/tex]
Therefore, the solution is:
[tex]\[ x = 2.297111188608965 \][/tex]
In summary, the solutions for [tex]\( x \)[/tex] are:
1. [tex]\( x = 6.900655593423542 \quad \text{or} \quad x = -6.900655593423542 \)[/tex]
2. [tex]\( x = 2.4264275032025866 \)[/tex]
3. [tex]\( x = 4.538816846833403 \quad \text{or} \quad x = -4.538816846833403 \)[/tex]
4. [tex]\( x = 1.2909944487358056 \quad \text{or} \quad x = -1.2909944487358056 \)[/tex]
5. [tex]\( x = 2.2104188991842317 \)[/tex]
6. [tex]\( x = 2.297111188608965 \)[/tex]
