Answer :
Sure, let's solve each equation by factoring step-by-step.
### 1. [tex]\( x^2 + 7x = 0 \)[/tex]
Factor out the common term [tex]\( x \)[/tex]:
[tex]\[ x(x + 7) = 0 \][/tex]
Set each factor to 0:
[tex]\[ x = 0 \][/tex]
[tex]\[ x + 7 = 0 \][/tex]
[tex]\[ x = -7 \][/tex]
So the solutions are [tex]\( x = 0 \)[/tex] and [tex]\( x = -7 \)[/tex].
### 2. [tex]\( 6s^2 + 18s = 0 \)[/tex]
Factor out the common term [tex]\( 6s \)[/tex]:
[tex]\[ 6s(s + 3) = 0 \][/tex]
Set each factor to 0:
[tex]\[ 6s = 0 \][/tex]
[tex]\[ s = 0 \][/tex]
[tex]\[ s + 3 = 0 \][/tex]
[tex]\[ s = -3 \][/tex]
So the solutions are [tex]\( s = 0 \)[/tex] and [tex]\( s = -3 \)[/tex].
### 3. [tex]\( t^2 + 8t + 16 = 0 \)[/tex]
This is a perfect square trinomial:
[tex]\[ t^2 + 2 \cdot 4 \cdot t + 4^2 = (t + 4)^2 = 0 \][/tex]
Set the factor to 0:
[tex]\[ (t + 4)^2 = 0 \][/tex]
[tex]\[ t + 4 = 0 \][/tex]
[tex]\[ t = -4 \][/tex]
So the solution is [tex]\( t = -4 \)[/tex].
### 4. [tex]\( x^2 - 10x + 25 = 0 \)[/tex]
This is a perfect square trinomial:
[tex]\[ x^2 - 2 \cdot 5 \cdot x + 5^2 = (x - 5)^2 = 0 \][/tex]
Set the factor to 0:
[tex]\[ (x - 5)^2 = 0 \][/tex]
[tex]\[ x - 5 = 0 \][/tex]
[tex]\[ x = 5 \][/tex]
So the solution is [tex]\( x = 5 \)[/tex].
### 5. [tex]\( h^2 + 6h = 16 \)[/tex]
Rewrite the equation in standard form:
[tex]\[ h^2 + 6h - 16 = 0 \][/tex]
Now, factor the trinomial:
[tex]\[ (h + 8)(h - 2) = 0 \][/tex]
Set each factor to 0:
[tex]\[ h + 8 = 0 \][/tex]
[tex]\[ h = -8 \][/tex]
[tex]\[ h - 2 = 0 \][/tex]
[tex]\[ h = 2 \][/tex]
So the solutions are [tex]\( h = -8 \)[/tex] and [tex]\( h = 2 \)[/tex].
### 6. [tex]\( x^2 - 14 = 5x \)[/tex]
Rewrite the equation in standard form:
[tex]\[ x^2 - 5x - 14 = 0 \][/tex]
Now, factor the trinomial:
[tex]\[ (x - 7)(x + 2) = 0 \][/tex]
Set each factor to 0:
[tex]\[ x - 7 = 0 \][/tex]
[tex]\[ x = 7 \][/tex]
[tex]\[ x + 2 = 0 \][/tex]
[tex]\[ x = -2 \][/tex]
So the solutions are [tex]\( x = 7 \)[/tex] and [tex]\( x = -2 \)[/tex].
### 7. [tex]\( 11r + 15 = -2r^2 \)[/tex]
Rewrite the equation in standard form:
[tex]\[ 2r^2 + 11r + 15 = 0 \][/tex]
Now, factor the trinomial:
[tex]\[ (2r + 5)(r + 3) = 0 \][/tex]
Set each factor to 0:
[tex]\[ 2r + 5 = 0 \][/tex]
[tex]\[ r = -\frac{5}{2} \][/tex]
[tex]\[ r + 3 = 0 \][/tex]
[tex]\[ r = -3 \][/tex]
So the solutions are [tex]\( r = -\frac{5}{2} \)[/tex] and [tex]\( r = -3 \)[/tex].
### 8. [tex]\( x^2 - 25 = 0 \)[/tex]
This is a difference of squares:
[tex]\[ x^2 - 5^2 = (x - 5)(x + 5) = 0 \][/tex]
Set each factor to 0:
[tex]\[ x - 5 = 0 \][/tex]
[tex]\[ x = 5 \][/tex]
[tex]\[ x + 5 = 0 \][/tex]
[tex]\[ x = -5 \][/tex]
So the solutions are [tex]\( x = 5 \)[/tex] and [tex]\( x = -5 \)[/tex].
### 9. [tex]\( 81 - 4x^2 = 0 \)[/tex]
Rewrite the equation:
[tex]\[ 81 = 4x^2 \][/tex]
Divide by 4:
[tex]\[ \frac{81}{4} = x^2 \][/tex]
Take the square root of both sides:
[tex]\[ x = \pm \frac{9}{2} \][/tex]
So the solutions are [tex]\( x = \frac{9}{2} \)[/tex] and [tex]\( x = -\frac{9}{2} \)[/tex].
### 10. [tex]\( 4s^2 + 9 = 12s \)[/tex]
Rewrite the equation in standard form:
[tex]\[ 4s^2 - 12s + 9 = 0 \][/tex]
Now, factor the trinomial:
[tex]\[ (2s - 3)^2 = 0 \][/tex]
Set the factor to 0:
[tex]\[ 2s - 3 = 0 \][/tex]
[tex]\[ s = \frac{3}{2} \][/tex]
So the solution is [tex]\( s = \frac{3}{2} \)[/tex].
By factoring these quadratic equations, we were able to determine the solutions for each.
### 1. [tex]\( x^2 + 7x = 0 \)[/tex]
Factor out the common term [tex]\( x \)[/tex]:
[tex]\[ x(x + 7) = 0 \][/tex]
Set each factor to 0:
[tex]\[ x = 0 \][/tex]
[tex]\[ x + 7 = 0 \][/tex]
[tex]\[ x = -7 \][/tex]
So the solutions are [tex]\( x = 0 \)[/tex] and [tex]\( x = -7 \)[/tex].
### 2. [tex]\( 6s^2 + 18s = 0 \)[/tex]
Factor out the common term [tex]\( 6s \)[/tex]:
[tex]\[ 6s(s + 3) = 0 \][/tex]
Set each factor to 0:
[tex]\[ 6s = 0 \][/tex]
[tex]\[ s = 0 \][/tex]
[tex]\[ s + 3 = 0 \][/tex]
[tex]\[ s = -3 \][/tex]
So the solutions are [tex]\( s = 0 \)[/tex] and [tex]\( s = -3 \)[/tex].
### 3. [tex]\( t^2 + 8t + 16 = 0 \)[/tex]
This is a perfect square trinomial:
[tex]\[ t^2 + 2 \cdot 4 \cdot t + 4^2 = (t + 4)^2 = 0 \][/tex]
Set the factor to 0:
[tex]\[ (t + 4)^2 = 0 \][/tex]
[tex]\[ t + 4 = 0 \][/tex]
[tex]\[ t = -4 \][/tex]
So the solution is [tex]\( t = -4 \)[/tex].
### 4. [tex]\( x^2 - 10x + 25 = 0 \)[/tex]
This is a perfect square trinomial:
[tex]\[ x^2 - 2 \cdot 5 \cdot x + 5^2 = (x - 5)^2 = 0 \][/tex]
Set the factor to 0:
[tex]\[ (x - 5)^2 = 0 \][/tex]
[tex]\[ x - 5 = 0 \][/tex]
[tex]\[ x = 5 \][/tex]
So the solution is [tex]\( x = 5 \)[/tex].
### 5. [tex]\( h^2 + 6h = 16 \)[/tex]
Rewrite the equation in standard form:
[tex]\[ h^2 + 6h - 16 = 0 \][/tex]
Now, factor the trinomial:
[tex]\[ (h + 8)(h - 2) = 0 \][/tex]
Set each factor to 0:
[tex]\[ h + 8 = 0 \][/tex]
[tex]\[ h = -8 \][/tex]
[tex]\[ h - 2 = 0 \][/tex]
[tex]\[ h = 2 \][/tex]
So the solutions are [tex]\( h = -8 \)[/tex] and [tex]\( h = 2 \)[/tex].
### 6. [tex]\( x^2 - 14 = 5x \)[/tex]
Rewrite the equation in standard form:
[tex]\[ x^2 - 5x - 14 = 0 \][/tex]
Now, factor the trinomial:
[tex]\[ (x - 7)(x + 2) = 0 \][/tex]
Set each factor to 0:
[tex]\[ x - 7 = 0 \][/tex]
[tex]\[ x = 7 \][/tex]
[tex]\[ x + 2 = 0 \][/tex]
[tex]\[ x = -2 \][/tex]
So the solutions are [tex]\( x = 7 \)[/tex] and [tex]\( x = -2 \)[/tex].
### 7. [tex]\( 11r + 15 = -2r^2 \)[/tex]
Rewrite the equation in standard form:
[tex]\[ 2r^2 + 11r + 15 = 0 \][/tex]
Now, factor the trinomial:
[tex]\[ (2r + 5)(r + 3) = 0 \][/tex]
Set each factor to 0:
[tex]\[ 2r + 5 = 0 \][/tex]
[tex]\[ r = -\frac{5}{2} \][/tex]
[tex]\[ r + 3 = 0 \][/tex]
[tex]\[ r = -3 \][/tex]
So the solutions are [tex]\( r = -\frac{5}{2} \)[/tex] and [tex]\( r = -3 \)[/tex].
### 8. [tex]\( x^2 - 25 = 0 \)[/tex]
This is a difference of squares:
[tex]\[ x^2 - 5^2 = (x - 5)(x + 5) = 0 \][/tex]
Set each factor to 0:
[tex]\[ x - 5 = 0 \][/tex]
[tex]\[ x = 5 \][/tex]
[tex]\[ x + 5 = 0 \][/tex]
[tex]\[ x = -5 \][/tex]
So the solutions are [tex]\( x = 5 \)[/tex] and [tex]\( x = -5 \)[/tex].
### 9. [tex]\( 81 - 4x^2 = 0 \)[/tex]
Rewrite the equation:
[tex]\[ 81 = 4x^2 \][/tex]
Divide by 4:
[tex]\[ \frac{81}{4} = x^2 \][/tex]
Take the square root of both sides:
[tex]\[ x = \pm \frac{9}{2} \][/tex]
So the solutions are [tex]\( x = \frac{9}{2} \)[/tex] and [tex]\( x = -\frac{9}{2} \)[/tex].
### 10. [tex]\( 4s^2 + 9 = 12s \)[/tex]
Rewrite the equation in standard form:
[tex]\[ 4s^2 - 12s + 9 = 0 \][/tex]
Now, factor the trinomial:
[tex]\[ (2s - 3)^2 = 0 \][/tex]
Set the factor to 0:
[tex]\[ 2s - 3 = 0 \][/tex]
[tex]\[ s = \frac{3}{2} \][/tex]
So the solution is [tex]\( s = \frac{3}{2} \)[/tex].
By factoring these quadratic equations, we were able to determine the solutions for each.
