Answer :
Sure, let's solve each of the given quadratic equations by extracting square roots step-by-step.
### 1. [tex]\( x^2 = 16 \)[/tex]
To solve for [tex]\( x \)[/tex], take the square root of both sides:
[tex]\[ x = \pm \sqrt{16} \][/tex]
[tex]\[ x = \pm 4 \][/tex]
Thus, the solutions are:
[tex]\[ x = 4 \quad \text{and} \quad x = -4 \][/tex]
### 2. [tex]\( t^2 = 81 \)[/tex]
To solve for [tex]\( t \)[/tex], take the square root of both sides:
[tex]\[ t = \pm \sqrt{81} \][/tex]
[tex]\[ t = \pm 9 \][/tex]
Thus, the solutions are:
[tex]\[ t = 9 \quad \text{and} \quad t = -9 \][/tex]
### 3. [tex]\( r^2 - 100 = 0 \)[/tex]
First, isolate the square term by adding 100 to both sides:
[tex]\[ r^2 = 100 \][/tex]
Now, take the square root of both sides:
[tex]\[ r = \pm \sqrt{100} \][/tex]
[tex]\[ r = \pm 10 \][/tex]
Thus, the solutions are:
[tex]\[ r = 10 \quad \text{and} \quad r = -10 \][/tex]
### 4. [tex]\( 4 x^2 - 144 = 0 \)[/tex]
First, isolate the square term by adding 144 to both sides:
[tex]\[ 4 x^2 = 144 \][/tex]
Next, divide both sides by 4:
[tex]\[ x^2 = 36 \][/tex]
Now, take the square root of both sides:
[tex]\[ x = \pm \sqrt{36} \][/tex]
[tex]\[ x = \pm 6 \][/tex]
Thus, the solutions are:
[tex]\[ x = 6 \quad \text{and} \quad x = -6 \][/tex]
### 5. [tex]\( 2 s^2 = 50 \)[/tex]
First, isolate the square term by dividing both sides by 2:
[tex]\[ s^2 = 25 \][/tex]
Now, take the square root of both sides:
[tex]\[ s = \pm \sqrt{25} \][/tex]
[tex]\[ s = \pm 5 \][/tex]
Thus, the solutions are:
[tex]\[ s = 5 \quad \text{and} \quad s = -5 \][/tex]
### 6. [tex]\( 4 x^2 - 225 = 0 \)[/tex]
First, isolate the square term by adding 225 to both sides:
[tex]\[ 4 x^2 = 225 \][/tex]
Next, divide both sides by 4:
[tex]\[ x^2 = 56.25 \][/tex]
Now, take the square root of both sides:
[tex]\[ x = \pm \sqrt{56.25} \][/tex]
[tex]\[ x = \pm 7.5 \][/tex]
Thus, the solutions are:
[tex]\[ x = 7.5 \quad \text{and} \quad x = -7.5 \][/tex]
### 7. [tex]\( 3 h^2 - 147 = 0 \)[/tex]
First, isolate the square term by adding 147 to both sides:
[tex]\[ 3 h^2 = 147 \][/tex]
Next, divide both sides by 3:
[tex]\[ h^2 = 49 \][/tex]
Now, take the square root of both sides:
[tex]\[ h = \pm \sqrt{49} \][/tex]
[tex]\[ h = \pm 7 \][/tex]
Thus, the solutions are:
[tex]\[ h = 7 \quad \text{and} \quad h = -7 \][/tex]
### 8. [tex]\( (x - 4)^2 = 169 \)[/tex]
To solve for [tex]\( x \)[/tex], take the square root of both sides:
[tex]\[ x - 4 = \pm \sqrt{169} \][/tex]
[tex]\[ x - 4 = \pm 13 \][/tex]
Now, solve for [tex]\( x \)[/tex]:
[tex]\[ x - 4 = 13 \quad \Rightarrow \quad x = 17 \][/tex]
[tex]\[ x - 4 = -13 \quad \Rightarrow \quad x = -9 \][/tex]
Thus, the solutions are:
[tex]\[ x = 17 \quad \text{and} \quad x = -9 \][/tex]
### 9. [tex]\( (k + 7)^2 = 289 \)[/tex]
To solve for [tex]\( k \)[/tex], take the square root of both sides:
[tex]\[ k + 7 = \pm \sqrt{289} \][/tex]
[tex]\[ k + 7 = \pm 17 \][/tex]
Now, solve for [tex]\( k \)[/tex]:
[tex]\[ k + 7 = 17 \quad \Rightarrow \quad k = 10 \][/tex]
[tex]\[ k + 7 = -17 \quad \Rightarrow \quad k = -24 \][/tex]
Thus, the solutions are:
[tex]\[ k = 10 \quad \text{and} \quad k = -24 \][/tex]
### 10. [tex]\( (2 s - 1)^2 = 225 \)[/tex]
To solve for [tex]\( s \)[/tex], take the square root of both sides:
[tex]\[ 2 s - 1 = \pm \sqrt{225} \][/tex]
[tex]\[ 2 s - 1 = \pm 15 \][/tex]
Now, solve for [tex]\( s \)[/tex]:
[tex]\[ 2 s - 1 = 15 \quad \Rightarrow \quad 2 s = 16 \quad \Rightarrow \quad s = 8 \][/tex]
[tex]\[ 2 s - 1 = -15 \quad \Rightarrow \quad 2 s = -14 \quad \Rightarrow \quad s = -7 \][/tex]
Thus, the solutions are:
[tex]\[ s = 8 \quad \text{and} \quad s = -7 \][/tex]
### 1. [tex]\( x^2 = 16 \)[/tex]
To solve for [tex]\( x \)[/tex], take the square root of both sides:
[tex]\[ x = \pm \sqrt{16} \][/tex]
[tex]\[ x = \pm 4 \][/tex]
Thus, the solutions are:
[tex]\[ x = 4 \quad \text{and} \quad x = -4 \][/tex]
### 2. [tex]\( t^2 = 81 \)[/tex]
To solve for [tex]\( t \)[/tex], take the square root of both sides:
[tex]\[ t = \pm \sqrt{81} \][/tex]
[tex]\[ t = \pm 9 \][/tex]
Thus, the solutions are:
[tex]\[ t = 9 \quad \text{and} \quad t = -9 \][/tex]
### 3. [tex]\( r^2 - 100 = 0 \)[/tex]
First, isolate the square term by adding 100 to both sides:
[tex]\[ r^2 = 100 \][/tex]
Now, take the square root of both sides:
[tex]\[ r = \pm \sqrt{100} \][/tex]
[tex]\[ r = \pm 10 \][/tex]
Thus, the solutions are:
[tex]\[ r = 10 \quad \text{and} \quad r = -10 \][/tex]
### 4. [tex]\( 4 x^2 - 144 = 0 \)[/tex]
First, isolate the square term by adding 144 to both sides:
[tex]\[ 4 x^2 = 144 \][/tex]
Next, divide both sides by 4:
[tex]\[ x^2 = 36 \][/tex]
Now, take the square root of both sides:
[tex]\[ x = \pm \sqrt{36} \][/tex]
[tex]\[ x = \pm 6 \][/tex]
Thus, the solutions are:
[tex]\[ x = 6 \quad \text{and} \quad x = -6 \][/tex]
### 5. [tex]\( 2 s^2 = 50 \)[/tex]
First, isolate the square term by dividing both sides by 2:
[tex]\[ s^2 = 25 \][/tex]
Now, take the square root of both sides:
[tex]\[ s = \pm \sqrt{25} \][/tex]
[tex]\[ s = \pm 5 \][/tex]
Thus, the solutions are:
[tex]\[ s = 5 \quad \text{and} \quad s = -5 \][/tex]
### 6. [tex]\( 4 x^2 - 225 = 0 \)[/tex]
First, isolate the square term by adding 225 to both sides:
[tex]\[ 4 x^2 = 225 \][/tex]
Next, divide both sides by 4:
[tex]\[ x^2 = 56.25 \][/tex]
Now, take the square root of both sides:
[tex]\[ x = \pm \sqrt{56.25} \][/tex]
[tex]\[ x = \pm 7.5 \][/tex]
Thus, the solutions are:
[tex]\[ x = 7.5 \quad \text{and} \quad x = -7.5 \][/tex]
### 7. [tex]\( 3 h^2 - 147 = 0 \)[/tex]
First, isolate the square term by adding 147 to both sides:
[tex]\[ 3 h^2 = 147 \][/tex]
Next, divide both sides by 3:
[tex]\[ h^2 = 49 \][/tex]
Now, take the square root of both sides:
[tex]\[ h = \pm \sqrt{49} \][/tex]
[tex]\[ h = \pm 7 \][/tex]
Thus, the solutions are:
[tex]\[ h = 7 \quad \text{and} \quad h = -7 \][/tex]
### 8. [tex]\( (x - 4)^2 = 169 \)[/tex]
To solve for [tex]\( x \)[/tex], take the square root of both sides:
[tex]\[ x - 4 = \pm \sqrt{169} \][/tex]
[tex]\[ x - 4 = \pm 13 \][/tex]
Now, solve for [tex]\( x \)[/tex]:
[tex]\[ x - 4 = 13 \quad \Rightarrow \quad x = 17 \][/tex]
[tex]\[ x - 4 = -13 \quad \Rightarrow \quad x = -9 \][/tex]
Thus, the solutions are:
[tex]\[ x = 17 \quad \text{and} \quad x = -9 \][/tex]
### 9. [tex]\( (k + 7)^2 = 289 \)[/tex]
To solve for [tex]\( k \)[/tex], take the square root of both sides:
[tex]\[ k + 7 = \pm \sqrt{289} \][/tex]
[tex]\[ k + 7 = \pm 17 \][/tex]
Now, solve for [tex]\( k \)[/tex]:
[tex]\[ k + 7 = 17 \quad \Rightarrow \quad k = 10 \][/tex]
[tex]\[ k + 7 = -17 \quad \Rightarrow \quad k = -24 \][/tex]
Thus, the solutions are:
[tex]\[ k = 10 \quad \text{and} \quad k = -24 \][/tex]
### 10. [tex]\( (2 s - 1)^2 = 225 \)[/tex]
To solve for [tex]\( s \)[/tex], take the square root of both sides:
[tex]\[ 2 s - 1 = \pm \sqrt{225} \][/tex]
[tex]\[ 2 s - 1 = \pm 15 \][/tex]
Now, solve for [tex]\( s \)[/tex]:
[tex]\[ 2 s - 1 = 15 \quad \Rightarrow \quad 2 s = 16 \quad \Rightarrow \quad s = 8 \][/tex]
[tex]\[ 2 s - 1 = -15 \quad \Rightarrow \quad 2 s = -14 \quad \Rightarrow \quad s = -7 \][/tex]
Thus, the solutions are:
[tex]\[ s = 8 \quad \text{and} \quad s = -7 \][/tex]
