Sure, let's solve each of the equations step-by-step by extracting the roots.
1. Solve the equation [tex]\(3t^2 = 12\)[/tex]
First, divide both sides of the equation by 3:
[tex]\[ t^2 = \frac{12}{3} \][/tex]
[tex]\[ t^2 = 4 \][/tex]
Next, take the square root of both sides of the equation:
[tex]\[ t = \sqrt{4} \][/tex]
Since the square root of 4 can be both positive and negative, we have two solutions:
[tex]\[ t = 2 \quad \text{or} \quad t = -2 \][/tex]
Thus, the solutions are:
[tex]\[ t = \{2, -2\} \][/tex]
2. Solve the equation [tex]\(2x^2 - 7 = 10\)[/tex]
First, add 7 to both sides of the equation:
[tex]\[ 2x^2 = 10 + 7 \][/tex]
[tex]\[ 2x^2 = 17 \][/tex]
Next, divide both sides by 2:
[tex]\[ x^2 = \frac{17}{2} \][/tex]
[tex]\[ x^2 = 8.5 \][/tex]
Then, take the square root of both sides:
[tex]\[ x = \sqrt{8.5} \][/tex]
Similarly, considering both positive and negative roots, the solutions are:
[tex]\[ x = \sqrt{8.5} \][/tex]
[tex]\[ x = -\sqrt{8.5} \][/tex]
So, the solutions are approximately:
[tex]\[ x = \{2.915, -2.915\} \][/tex]
3. Solve the equation [tex]\(x^2 = 50\)[/tex]
Take the square root of both sides:
[tex]\[ x = \sqrt{50} \][/tex]
Considering both positive and negative roots, we get:
[tex]\[ x = \sqrt{50} \][/tex]
[tex]\[ x = -\sqrt{50} \][/tex]
So, the solutions are approximately:
[tex]\[ x = \{7.071, -7.071\} \][/tex]
4. Solve the equation [tex]\(4.3r^3 = 18\)[/tex]
First, divide both sides of the equation by 4.3:
[tex]\[ r^3 = \frac{18}{4.3} \][/tex]
[tex]\[ r^3 \approx 4.186 \][/tex]
Next, take the cube root of both sides:
[tex]\[ r = \sqrt[3]{4.186} \][/tex]
Considering both positive and negative roots, we get:
[tex]\[ r = \sqrt[3]{4.186} \][/tex]
[tex]\[ r = -\sqrt[3]{4.186} \][/tex]
So, the solutions are approximately:
[tex]\[ r = \{1.612, -1.612\} \][/tex]
5. Solve the equation [tex]\((5-4)^2 - 81 = 0\)[/tex]
First, simplify inside the parentheses:
[tex]\[ (1)^2 - 81 = 0 \][/tex]
[tex]\[ 1 - 81 = 0 \][/tex]
[tex]\[ -80 = 0 \][/tex]
Since [tex]\(-80\)[/tex] does not equal 0, this equation has no solution.
Thus, the solutions are:
1. [tex]\(t = \{2, -2\}\)[/tex]
2. [tex]\(x = \{2.915, -2.915\}\)[/tex]
3. [tex]\(x = \{7.071, -7.071\}\)[/tex]
4. [tex]\(r = \{1.612, -1.612\}\)[/tex]
5. No solution
