Let's solve each equation step by step.
### 1. Solve the equation [tex]\( x^2 - 7 = 0 \)[/tex]
The equation given is:
[tex]\[ x^2 - 7 = 0 \][/tex]
1.1 Isolate [tex]\( x^2 \)[/tex] by adding 7 to both sides:
[tex]\[ x^2 = 7 \][/tex]
1.2 Take the square root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \pm \sqrt{7} \][/tex]
Thus, the solutions for this equation are:
[tex]\[ x = \sqrt{7} \][/tex] and [tex]\[ x = -\sqrt{7} \][/tex]
### 2. Solve the equation [tex]\( 3 \cdot 3t^2 = 18 \)[/tex]
The equation given is:
[tex]\[ 3 \cdot 3t^2 = 18 \][/tex]
2.1 Simplify the left side of the equation:
[tex]\[ 9t^2 = 18 \][/tex]
2.2 Divide both sides by 9 to isolate [tex]\( t^2 \)[/tex]:
[tex]\[ t^2 = 2 \][/tex]
2.3 Take the square root of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[ t = \pm \sqrt{2} \][/tex]
Thus, the solutions for this equation are:
[tex]\[ t = \sqrt{2} \][/tex] and [tex]\[ t = -\sqrt{2} \][/tex]
### 3. Solve the equation [tex]\( 4 \cdot (5 - 4)^2 - 81 = 0 \)[/tex]
The equation given is:
[tex]\[ 4 \cdot (5 - 4)^2 - 81 = 0 \][/tex]
3.1 Simplify the expression inside the parentheses:
[tex]\[ 4 \cdot (1)^2 - 81 = 0 \][/tex]
3.2 Simplify further:
[tex]\[ 4 \cdot 1 - 81 = 0 \][/tex]
3.3 Perform the multiplication:
[tex]\[ 4 - 81 = 0 \][/tex]
3.4 Simplify the equation:
[tex]\[ -77 = 0 \][/tex]
The simplified form is:
[tex]\[ -77 \][/tex]
This means that the equation does not hold true as it stands and thus has no solution. The result -77 indicates the equation setup does not equal zero, thus the given equation is inconsistent.
In summary, the solutions are:
1. For equation [tex]\( x^2 - 7 = 0 \)[/tex]:
[tex]\[ x = \sqrt{7}, \; x = -\sqrt{7} \][/tex]
2. For equation [tex]\( 3 \cdot 3t^2 = 18 \)[/tex]:
[tex]\[ t = \sqrt{2}, \; t = -\sqrt{2} \][/tex]
3. For equation [tex]\( 4 \cdot (5 - 4)^2 - 81 = 0 \)[/tex]:
[tex]\[ \text{This equation has no solution, it simplifies to } -77 \][/tex]
