Answer :
Sure, let's solve each of these quadratic equations by factoring step by step.
### 1. Solve [tex]\( x^2 + 7x = 0 \)[/tex]
We can factor out the common term [tex]\(x\)[/tex]:
[tex]\[ x(x + 7) = 0 \][/tex]
Set each factor to zero:
[tex]\[ x = 0 \][/tex]
[tex]\[ x + 7 = 0 \Rightarrow x = -7 \][/tex]
So, the solutions are:
[tex]\[ x = 0, -7 \][/tex]
### 2. Solve [tex]\( 6s^2 + 18s = 0 \)[/tex]
We can factor out the common term [tex]\(6s\)[/tex]:
[tex]\[ 6s(s + 3) = 0 \][/tex]
Set each factor to zero:
[tex]\[ 6s = 0 \Rightarrow s = 0 \][/tex]
[tex]\[ s + 3 = 0 \Rightarrow s = -3 \][/tex]
So, the solutions are:
[tex]\[ s = 0, -3 \][/tex]
### 6. Solve [tex]\( x^2 - 14 = 5x \)[/tex]
First, rearrange the equation to set it to zero:
[tex]\[ x^2 - 5x - 14 = 0 \][/tex]
Now, factor the quadratic:
[tex]\[ (x - 7)(x + 2) = 0 \][/tex]
Set each factor to zero:
[tex]\[ x - 7 = 0 \Rightarrow x = 7 \][/tex]
[tex]\[ x + 2 = 0 \Rightarrow x = -2 \][/tex]
So, the solutions are:
[tex]\[ x = 7, -2 \][/tex]
### 7. Solve [tex]\( 11r + 15 = -2r \)[/tex]
First, rearrange the equation to set it to zero:
[tex]\[ 11r + 15 + 2r = 0 \Rightarrow 13r + 15 = 0 \][/tex]
Now, solve for [tex]\(r\)[/tex]:
[tex]\[ 13r = -15 \Rightarrow r = -\frac{15}{13} \][/tex]
So, the solution is:
[tex]\[ r = -\frac{15}{13} \][/tex]
### 3. Solve [tex]\( t^2 + 8t + 16 = 0 \)[/tex]
This equation is a perfect square:
[tex]\[ (t + 4)^2 = 0 \][/tex]
Set the factor to zero:
[tex]\[ t + 4 = 0 \Rightarrow t = -4 \][/tex]
So, the solution is:
[tex]\[ t = -4 \][/tex]
### 8. Solve [tex]\( x^2 - 25 = 0 \)[/tex]
This equation is a difference of squares:
[tex]\[ (x - 5)(x + 5) = 0 \][/tex]
Set each factor to zero:
[tex]\[ x - 5 = 0 \Rightarrow x = 5 \][/tex]
[tex]\[ x + 5 = 0 \Rightarrow x = -5 \][/tex]
So, the solutions are:
[tex]\[ x = 5, -5 \][/tex]
### 4. Solve [tex]\( x^2 - 10x + 25 = 0 \)[/tex]
This equation is a perfect square:
[tex]\[ (x - 5)^2 = 0 \][/tex]
Set the factor to zero:
[tex]\[ x - 5 = 0 \Rightarrow x = 5 \][/tex]
So, the solution is:
[tex]\[ x = 5 \][/tex]
### 9. Solve [tex]\( 81 - 4x^2 = 0 \)[/tex]
This equation is a difference of squares:
[tex]\[ (9 - 2x)(9 + 2x) = 0 \][/tex]
Set each factor to zero:
[tex]\[9 - 2x = 0 \Rightarrow 2x = 9 \Rightarrow x = \frac{9}{2} \][/tex]
[tex]\[ 9 + 2x = 0 \Rightarrow 2x = -9 \Rightarrow x = -\frac{9}{2} \][/tex]
So, the solutions are:
[tex]\[ x = \frac{9}{2}, -\frac{9}{2} \][/tex]
### 5. Solve [tex]\( h^2 + 6h = 16 \)[/tex]
First, rearrange the equation to set it to zero:
[tex]\[ h^2 + 6h - 16 = 0 \][/tex]
Now, factor the quadratic:
[tex]\[ (h + 8)(h - 2) = 0 \][/tex]
Set each factor to zero:
[tex]\[ h + 8 = 0 \Rightarrow h = -8 \][/tex]
[tex]\[ h - 2 = 0 \Rightarrow h = 2 \][/tex]
So, the solutions are:
[tex]\[ h = -8, 2 \][/tex]
### 10. Solve [tex]\( 45x^2 + 9 = 123 \)[/tex]
First, rearrange the equation to set it to zero:
[tex]\[ 45x^2 + 9 - 123 = 0 \Rightarrow 45x^2 - 114 = 0 \][/tex]
Factor out the common term:
[tex]\[ 45x^2 - 114 = 0 \Rightarrow x = \pm \sqrt{\frac{114}{45}} \][/tex]
Simplify the fraction:
[tex]\[ x = \pm \sqrt{\frac{38}{15}} \Rightarrow x = \pm \frac{\sqrt{570}}{15} \][/tex]
So, the solutions are:
[tex]\[ x = \frac{\sqrt{570}}{15}, -\frac{\sqrt{570}}{15} \][/tex]
### 1. Solve [tex]\( x^2 + 7x = 0 \)[/tex]
We can factor out the common term [tex]\(x\)[/tex]:
[tex]\[ x(x + 7) = 0 \][/tex]
Set each factor to zero:
[tex]\[ x = 0 \][/tex]
[tex]\[ x + 7 = 0 \Rightarrow x = -7 \][/tex]
So, the solutions are:
[tex]\[ x = 0, -7 \][/tex]
### 2. Solve [tex]\( 6s^2 + 18s = 0 \)[/tex]
We can factor out the common term [tex]\(6s\)[/tex]:
[tex]\[ 6s(s + 3) = 0 \][/tex]
Set each factor to zero:
[tex]\[ 6s = 0 \Rightarrow s = 0 \][/tex]
[tex]\[ s + 3 = 0 \Rightarrow s = -3 \][/tex]
So, the solutions are:
[tex]\[ s = 0, -3 \][/tex]
### 6. Solve [tex]\( x^2 - 14 = 5x \)[/tex]
First, rearrange the equation to set it to zero:
[tex]\[ x^2 - 5x - 14 = 0 \][/tex]
Now, factor the quadratic:
[tex]\[ (x - 7)(x + 2) = 0 \][/tex]
Set each factor to zero:
[tex]\[ x - 7 = 0 \Rightarrow x = 7 \][/tex]
[tex]\[ x + 2 = 0 \Rightarrow x = -2 \][/tex]
So, the solutions are:
[tex]\[ x = 7, -2 \][/tex]
### 7. Solve [tex]\( 11r + 15 = -2r \)[/tex]
First, rearrange the equation to set it to zero:
[tex]\[ 11r + 15 + 2r = 0 \Rightarrow 13r + 15 = 0 \][/tex]
Now, solve for [tex]\(r\)[/tex]:
[tex]\[ 13r = -15 \Rightarrow r = -\frac{15}{13} \][/tex]
So, the solution is:
[tex]\[ r = -\frac{15}{13} \][/tex]
### 3. Solve [tex]\( t^2 + 8t + 16 = 0 \)[/tex]
This equation is a perfect square:
[tex]\[ (t + 4)^2 = 0 \][/tex]
Set the factor to zero:
[tex]\[ t + 4 = 0 \Rightarrow t = -4 \][/tex]
So, the solution is:
[tex]\[ t = -4 \][/tex]
### 8. Solve [tex]\( x^2 - 25 = 0 \)[/tex]
This equation is a difference of squares:
[tex]\[ (x - 5)(x + 5) = 0 \][/tex]
Set each factor to zero:
[tex]\[ x - 5 = 0 \Rightarrow x = 5 \][/tex]
[tex]\[ x + 5 = 0 \Rightarrow x = -5 \][/tex]
So, the solutions are:
[tex]\[ x = 5, -5 \][/tex]
### 4. Solve [tex]\( x^2 - 10x + 25 = 0 \)[/tex]
This equation is a perfect square:
[tex]\[ (x - 5)^2 = 0 \][/tex]
Set the factor to zero:
[tex]\[ x - 5 = 0 \Rightarrow x = 5 \][/tex]
So, the solution is:
[tex]\[ x = 5 \][/tex]
### 9. Solve [tex]\( 81 - 4x^2 = 0 \)[/tex]
This equation is a difference of squares:
[tex]\[ (9 - 2x)(9 + 2x) = 0 \][/tex]
Set each factor to zero:
[tex]\[9 - 2x = 0 \Rightarrow 2x = 9 \Rightarrow x = \frac{9}{2} \][/tex]
[tex]\[ 9 + 2x = 0 \Rightarrow 2x = -9 \Rightarrow x = -\frac{9}{2} \][/tex]
So, the solutions are:
[tex]\[ x = \frac{9}{2}, -\frac{9}{2} \][/tex]
### 5. Solve [tex]\( h^2 + 6h = 16 \)[/tex]
First, rearrange the equation to set it to zero:
[tex]\[ h^2 + 6h - 16 = 0 \][/tex]
Now, factor the quadratic:
[tex]\[ (h + 8)(h - 2) = 0 \][/tex]
Set each factor to zero:
[tex]\[ h + 8 = 0 \Rightarrow h = -8 \][/tex]
[tex]\[ h - 2 = 0 \Rightarrow h = 2 \][/tex]
So, the solutions are:
[tex]\[ h = -8, 2 \][/tex]
### 10. Solve [tex]\( 45x^2 + 9 = 123 \)[/tex]
First, rearrange the equation to set it to zero:
[tex]\[ 45x^2 + 9 - 123 = 0 \Rightarrow 45x^2 - 114 = 0 \][/tex]
Factor out the common term:
[tex]\[ 45x^2 - 114 = 0 \Rightarrow x = \pm \sqrt{\frac{114}{45}} \][/tex]
Simplify the fraction:
[tex]\[ x = \pm \sqrt{\frac{38}{15}} \Rightarrow x = \pm \frac{\sqrt{570}}{15} \][/tex]
So, the solutions are:
[tex]\[ x = \frac{\sqrt{570}}{15}, -\frac{\sqrt{570}}{15} \][/tex]
