Answer :
Para resolver las ecuaciones cuadráticas y lineales dadas, seguimos los siguientes pasos:
### Ecuaciones lineales
Resolveremos las ecuaciones lineales primero. Tenemos las siguientes ecuaciones:
1. [tex]\( x - 20 = 3 \)[/tex]
2. [tex]\( x + 9 = 31 \)[/tex]
3. [tex]\( 12x + 4 = 100 \)[/tex]
4. [tex]\( 10x - 12 = 28 \)[/tex]
#### 1. [tex]\( x - 20 = 3 \)[/tex]
[tex]\[ x - 20 = 3 \\ x = 3 + 20 \\ x = 23 \][/tex]
#### 2. [tex]\( x + 9 = 31 \)[/tex]
[tex]\[ x + 9 = 31 \\ x = 31 - 9 \\ x = 22 \][/tex]
#### 3. [tex]\( 12x + 4 = 100 \)[/tex]
[tex]\[ 12x + 4 = 100 \\ 12x = 100 - 4 \\ 12x = 96 \\ x = \frac{96}{12} \\ x = 8 \][/tex]
#### 4. [tex]\( 10x - 12 = 28 \)[/tex]
[tex]\[ 10x - 12 = 28 \\ 10x = 28 + 12 \\ 10x = 40 \\ x = \frac{40}{10} \\ x = 4 \][/tex]
### Ecuaciones de la forma [tex]\( ax + b = c \)[/tex]
Pasamos a resolver las ecuaciones de la forma [tex]\( ax + b = cx + d \)[/tex]:
1. [tex]\( 10x - 28 = -6x + 20 \)[/tex]
2. [tex]\( 2x - 34 = -2x + 2 \)[/tex]
3. [tex]\( 10x - 16 = -4x + 40 \)[/tex]
4. [tex]\( 24x + 80 = -16 + 200 \)[/tex]
5. [tex]\( 24x - 160 = -12x + 20 \)[/tex]
#### 1. [tex]\( 10x - 28 = -6x + 20 \)[/tex]
[tex]\[ 10x - 28 = -6x + 20 \\ 10x + 6x = 20 + 28 \\ 16x = 48 \\ x = \frac{48}{16} \\ x = 3 \][/tex]
#### 2. [tex]\( 2x - 34 = -2x + 2 \)[/tex]
[tex]\[ 2x - 34 = -2x + 2 \\ 2x + 2x = 2 + 34 \\ 4x = 36 \\ x = \frac{36}{4} \\ x = 9 \][/tex]
#### 3. [tex]\( 10x - 16 = -4x + 40 \)[/tex]
[tex]\[ 10x - 16 = -4x + 40 \\ 10x + 4x = 40 + 16 \\ 14x = 56 \\ x = \frac{56}{14} \\ x = 4 \][/tex]
#### 4. [tex]\( 24x + 80 = -16 + 200 \)[/tex]
[tex]\[ 24x + 80 = 184 \\ 24x = 104 \\ x = \frac{104}{24} \\ x = \frac{13}{3} \][/tex]
#### 5. [tex]\( 24x - 160 = -12x + 20 \)[/tex]
[tex]\[ 24x - 160 = -12x + 20 \\ 24x + 12x = 20 + 160 \\ 36x = 180 \\ x = \frac{180}{36} \\ x = 5 \][/tex]
### Ecuaciones cuadráticas de la forma [tex]\( ax^2 + c = 0 \)[/tex]
Ahora, resolveremos las ecuaciones cuadráticas en la forma [tex]\( ax^2 + c = 0 \)[/tex]:
1. [tex]\( 5x^2 - 720 = 0 \)[/tex]
2. [tex]\( 6x^2 - 96 = 0 \)[/tex]
3. [tex]\( 3x^2 - 147 = 0 \)[/tex]
4. [tex]\( 3x^2 - 75 = 0 \)[/tex]
#### 1. [tex]\( 5x^2 - 720 = 0 \)[/tex]
[tex]\[ 5x^2 - 720 = 0 \\ 5x^2 = 720 \\ x^2 = \frac{720}{5} \\ x^2 = 144 \\ x = \pm \sqrt{144} \\ x = \pm 12 \][/tex]
#### 2. [tex]\( 6x^2 - 96 = 0 \)[/tex]
[tex]\[ 6x^2 - 96 = 0 \\ 6x^2 = 96 \\ x^2 = \frac{96}{6} \\ x^2 = 16 \\ x = \pm \sqrt{16} \\ x = \pm 4 \][/tex]
#### 3. [tex]\( 3x^2 - 147 = 0 \)[/tex]
[tex]\[ 3x^2 - 147 = 0 \\ 3x^2 = 147 \\ x^2 = \frac{147}{3} \\ x^2 = 49 \\ x = \pm \sqrt{49} \\ x = \pm 7 \][/tex]
#### 4. [tex]\( 3x^2 - 75 = 0 \)[/tex]
[tex]\[ 3x^2 - 75 = 0 \\ 3x^2 = 75 \\ x^2 = \frac{75}{3} \\ x^2 = 25 \\ x = \pm \sqrt{25} \\ x = \pm 5 \][/tex]
### Ecuaciones cuadráticas de la forma [tex]\( ax^2 = c \)[/tex]
Finalmente, resolveremos las ecuaciones cuadráticas en la forma [tex]\( ax^2 - c = 0 \)[/tex]:
[tex]\[ 2x^2 - 3 = 95 \\ 6x^2 - 16 = 200 \\ 8x^2 + 2 = 34 \\ 7x^2 - 8 = 440 \\ 4x^2 + 4 = 148 \\ 3x^2 + 3 = 30 \][/tex]
#### 1. [tex]\( 2x^2 - 3 = 95 \)[/tex]
[tex]\[ 2x^2 - 3 = 95 \\ 2x^2 = 98 \\ x^2 = \frac{98}{2} \\ x^2 = 49 \\ x = \pm 7 \][/tex]
#### 2. [tex]\( 6x^2 - 16 = 200 \)[/tex]
[tex]\[ 6x^2 - 16 = 200 \\ 6x^2 = 216 \\ x^2 = \frac{216}{6} \\ x^2 = 36 \\ x = \pm 6 \][/tex]
#### 3. [tex]\( 8x^2 + 2 = 34 \)[/tex]
[tex]\[ 8x^2 + 2 = 34 \\ 8x^2 = 32 \\ x^2 = \frac{32}{8} \\ x^2 = 4 \\ x = \pm 2 \][/tex]
#### 4. [tex]\( 7x^2 - 8 = 440 \)[/tex]
[tex]\[ 7x^2 - 8 = 440 \\ 7x^2 = 448 \\ x^2 = \frac{448}{7} \\ x^2 = 64 \\ x = \pm 8 \][/tex]
#### 5. [tex]\( 4x^2 + 4 = 148 \)[/tex]
[tex]\[ 4x^2 + 4 = 148 \\ 4x^2 = 144 \\ x^2 = \frac{144}{4} \\ x^2 = 36 \\ x = \pm 6 \][/tex]
#### 6. [tex]\( 3x^2 + 3 = 30 \)[/tex]
[tex]\[ 3x^2 + 3 = 30 \\ 3x^2 = 27 \\ x^2 = \frac{27}{3} \\ x^2 = 9 \\ x = \pm 3 \][/tex]
Espero que esta explicación paso a paso te haya ayudado a comprender cómo resolver cada una de las ecuaciones proporcionadas. Si tienes más preguntas o necesitas alguna aclaración, ¡no dudes en preguntar!
### Ecuaciones lineales
Resolveremos las ecuaciones lineales primero. Tenemos las siguientes ecuaciones:
1. [tex]\( x - 20 = 3 \)[/tex]
2. [tex]\( x + 9 = 31 \)[/tex]
3. [tex]\( 12x + 4 = 100 \)[/tex]
4. [tex]\( 10x - 12 = 28 \)[/tex]
#### 1. [tex]\( x - 20 = 3 \)[/tex]
[tex]\[ x - 20 = 3 \\ x = 3 + 20 \\ x = 23 \][/tex]
#### 2. [tex]\( x + 9 = 31 \)[/tex]
[tex]\[ x + 9 = 31 \\ x = 31 - 9 \\ x = 22 \][/tex]
#### 3. [tex]\( 12x + 4 = 100 \)[/tex]
[tex]\[ 12x + 4 = 100 \\ 12x = 100 - 4 \\ 12x = 96 \\ x = \frac{96}{12} \\ x = 8 \][/tex]
#### 4. [tex]\( 10x - 12 = 28 \)[/tex]
[tex]\[ 10x - 12 = 28 \\ 10x = 28 + 12 \\ 10x = 40 \\ x = \frac{40}{10} \\ x = 4 \][/tex]
### Ecuaciones de la forma [tex]\( ax + b = c \)[/tex]
Pasamos a resolver las ecuaciones de la forma [tex]\( ax + b = cx + d \)[/tex]:
1. [tex]\( 10x - 28 = -6x + 20 \)[/tex]
2. [tex]\( 2x - 34 = -2x + 2 \)[/tex]
3. [tex]\( 10x - 16 = -4x + 40 \)[/tex]
4. [tex]\( 24x + 80 = -16 + 200 \)[/tex]
5. [tex]\( 24x - 160 = -12x + 20 \)[/tex]
#### 1. [tex]\( 10x - 28 = -6x + 20 \)[/tex]
[tex]\[ 10x - 28 = -6x + 20 \\ 10x + 6x = 20 + 28 \\ 16x = 48 \\ x = \frac{48}{16} \\ x = 3 \][/tex]
#### 2. [tex]\( 2x - 34 = -2x + 2 \)[/tex]
[tex]\[ 2x - 34 = -2x + 2 \\ 2x + 2x = 2 + 34 \\ 4x = 36 \\ x = \frac{36}{4} \\ x = 9 \][/tex]
#### 3. [tex]\( 10x - 16 = -4x + 40 \)[/tex]
[tex]\[ 10x - 16 = -4x + 40 \\ 10x + 4x = 40 + 16 \\ 14x = 56 \\ x = \frac{56}{14} \\ x = 4 \][/tex]
#### 4. [tex]\( 24x + 80 = -16 + 200 \)[/tex]
[tex]\[ 24x + 80 = 184 \\ 24x = 104 \\ x = \frac{104}{24} \\ x = \frac{13}{3} \][/tex]
#### 5. [tex]\( 24x - 160 = -12x + 20 \)[/tex]
[tex]\[ 24x - 160 = -12x + 20 \\ 24x + 12x = 20 + 160 \\ 36x = 180 \\ x = \frac{180}{36} \\ x = 5 \][/tex]
### Ecuaciones cuadráticas de la forma [tex]\( ax^2 + c = 0 \)[/tex]
Ahora, resolveremos las ecuaciones cuadráticas en la forma [tex]\( ax^2 + c = 0 \)[/tex]:
1. [tex]\( 5x^2 - 720 = 0 \)[/tex]
2. [tex]\( 6x^2 - 96 = 0 \)[/tex]
3. [tex]\( 3x^2 - 147 = 0 \)[/tex]
4. [tex]\( 3x^2 - 75 = 0 \)[/tex]
#### 1. [tex]\( 5x^2 - 720 = 0 \)[/tex]
[tex]\[ 5x^2 - 720 = 0 \\ 5x^2 = 720 \\ x^2 = \frac{720}{5} \\ x^2 = 144 \\ x = \pm \sqrt{144} \\ x = \pm 12 \][/tex]
#### 2. [tex]\( 6x^2 - 96 = 0 \)[/tex]
[tex]\[ 6x^2 - 96 = 0 \\ 6x^2 = 96 \\ x^2 = \frac{96}{6} \\ x^2 = 16 \\ x = \pm \sqrt{16} \\ x = \pm 4 \][/tex]
#### 3. [tex]\( 3x^2 - 147 = 0 \)[/tex]
[tex]\[ 3x^2 - 147 = 0 \\ 3x^2 = 147 \\ x^2 = \frac{147}{3} \\ x^2 = 49 \\ x = \pm \sqrt{49} \\ x = \pm 7 \][/tex]
#### 4. [tex]\( 3x^2 - 75 = 0 \)[/tex]
[tex]\[ 3x^2 - 75 = 0 \\ 3x^2 = 75 \\ x^2 = \frac{75}{3} \\ x^2 = 25 \\ x = \pm \sqrt{25} \\ x = \pm 5 \][/tex]
### Ecuaciones cuadráticas de la forma [tex]\( ax^2 = c \)[/tex]
Finalmente, resolveremos las ecuaciones cuadráticas en la forma [tex]\( ax^2 - c = 0 \)[/tex]:
[tex]\[ 2x^2 - 3 = 95 \\ 6x^2 - 16 = 200 \\ 8x^2 + 2 = 34 \\ 7x^2 - 8 = 440 \\ 4x^2 + 4 = 148 \\ 3x^2 + 3 = 30 \][/tex]
#### 1. [tex]\( 2x^2 - 3 = 95 \)[/tex]
[tex]\[ 2x^2 - 3 = 95 \\ 2x^2 = 98 \\ x^2 = \frac{98}{2} \\ x^2 = 49 \\ x = \pm 7 \][/tex]
#### 2. [tex]\( 6x^2 - 16 = 200 \)[/tex]
[tex]\[ 6x^2 - 16 = 200 \\ 6x^2 = 216 \\ x^2 = \frac{216}{6} \\ x^2 = 36 \\ x = \pm 6 \][/tex]
#### 3. [tex]\( 8x^2 + 2 = 34 \)[/tex]
[tex]\[ 8x^2 + 2 = 34 \\ 8x^2 = 32 \\ x^2 = \frac{32}{8} \\ x^2 = 4 \\ x = \pm 2 \][/tex]
#### 4. [tex]\( 7x^2 - 8 = 440 \)[/tex]
[tex]\[ 7x^2 - 8 = 440 \\ 7x^2 = 448 \\ x^2 = \frac{448}{7} \\ x^2 = 64 \\ x = \pm 8 \][/tex]
#### 5. [tex]\( 4x^2 + 4 = 148 \)[/tex]
[tex]\[ 4x^2 + 4 = 148 \\ 4x^2 = 144 \\ x^2 = \frac{144}{4} \\ x^2 = 36 \\ x = \pm 6 \][/tex]
#### 6. [tex]\( 3x^2 + 3 = 30 \)[/tex]
[tex]\[ 3x^2 + 3 = 30 \\ 3x^2 = 27 \\ x^2 = \frac{27}{3} \\ x^2 = 9 \\ x = \pm 3 \][/tex]
Espero que esta explicación paso a paso te haya ayudado a comprender cómo resolver cada una de las ecuaciones proporcionadas. Si tienes más preguntas o necesitas alguna aclaración, ¡no dudes en preguntar!
