Certainly! Let's solve each of the following equations step by step.
### 1. Solve [tex]\( 12x^2 = 48x \)[/tex]
1. Rewrite the equation:
[tex]\[ 12x^2 - 48x = 0 \][/tex]
2. Factor out the common term [tex]\( x \)[/tex]:
[tex]\[ x (12x - 48) = 0 \][/tex]
3. Set each factor to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x = 0 \quad \text{or} \quad 12x - 48 = 0 \][/tex]
[tex]\[ 12x - 48 = 0 \implies x = 4 \][/tex]
So the solutions are:
[tex]\[ x = 0 \ \text{or} \ x = 4 \][/tex]
### 2. Solve [tex]\( 3x - 12x^2 = 0 \)[/tex]
1. Rewrite the equation:
[tex]\[ -12x^2 + 3x = 0 \][/tex]
2. Factor out the common term [tex]\( x \)[/tex]:
[tex]\[ x (-12x + 3) = 0 \][/tex]
3. Set each factor to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x = 0 \quad \text{or} \quad -12x + 3 = 0 \][/tex]
[tex]\[ -12x + 3 = 0 \implies x = \frac{1}{4} \][/tex]
So the solutions are:
[tex]\[ x = 0 \ \text{or} \ x = \frac{1}{4} \][/tex]
### 3. Solve [tex]\( 8r^2 - 15r = 40 \)[/tex]
1. Rewrite the equation:
[tex]\[ 8r^2 - 15r - 40 = 0 \][/tex]
2. Use the quadratic formula [tex]\( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex] where [tex]\( a = 8 \)[/tex], [tex]\( b = -15 \)[/tex], and [tex]\( c = -40 \)[/tex]:
[tex]\[ r = \frac{-(-15) \pm \sqrt{(-15)^2 - 4 \cdot 8 \cdot (-40)}}{2 \cdot 8} \][/tex]
[tex]\[ r = \frac{15 \pm \sqrt{225 + 1280}}{16} \][/tex]
[tex]\[ r = \frac{15 \pm \sqrt{1505}}{16} \][/tex]
So the solutions are:
[tex]\[ r = \frac{15 + \sqrt{1505}}{16} \ \text{or} \ r = \frac{15 - \sqrt{1505}}{16} \][/tex]
### 4. Solve [tex]\( 120x - 12x^2 = 0 \)[/tex]
1. Rewrite the equation:
[tex]\[ -12x^2 + 120x = 0 \][/tex]
2. Factor out the common term [tex]\( x \)[/tex]:
[tex]\[ x (-12x + 120) = 0 \][/tex]
3. Set each factor to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x = 0 \quad \text{or} \quad -12x + 120 = 0 \][/tex]
[tex]\[ -12x + 120 = 0 \implies x = 10 \][/tex]
So the solutions are:
[tex]\[ x = 0 \ \text{or} \ x = 10 \][/tex]
### 5. Solve [tex]\( 210j^2 + 5j = -5j \)[/tex]
1. Rewrite the equation:
[tex]\[ 210j^2 + 5j + 5j = 0 \][/tex]
[tex]\[ 210j^2 + 10j = 0 \][/tex]
2. Factor out the common term [tex]\( j \)[/tex]:
[tex]\[ j (210j + 10) = 0 \][/tex]
3. Set each factor to zero and solve for [tex]\( j \)[/tex]:
[tex]\[ j = 0 \quad \text{or} \quad 210j + 10 = 0 \][/tex]
[tex]\[ 210j + 10 = 0 \implies j = -\frac{1}{21} \][/tex]
So the solutions are:
[tex]\[ j = 0 \ \text{or} \ j = -\frac{1}{21} \][/tex]
### Summary:
1. [tex]\( 12x^2 = 48x \)[/tex] gives [tex]\( x = 0 \ \text{or} \ x = 4 \)[/tex]
2. [tex]\( 3x - 12x^2 = 0 \)[/tex] gives [tex]\( x = 0 \ \text{or} \ x = \frac{1}{4} \)[/tex]
3. [tex]\( 8r^2 - 15r = 40 \)[/tex] gives [tex]\( r = \frac{15 - \sqrt{1505}}{16} \ \text{or} \ r = \frac{15 + \sqrt{1505}}{16} \)[/tex]
4. [tex]\( 120x - 12x^2 = 0 \)[/tex] gives [tex]\( x = 0 \ \text{or} \ x = 10 \)[/tex]
5. [tex]\( 210j^2 + 5j = -5j \)[/tex] gives [tex]\( j = 0 \ \text{or} \ j = -\frac{1}{21} \)[/tex]
