Answer :
To solve the quadratic equation [tex]\( 3(x+7)^2 - 32 = -29 \)[/tex], follow these steps:
1. Simplify the equation:
[tex]\[ 3(x+7)^2 - 32 = -29 \][/tex]
Add 32 to both sides to isolate the quadratic term:
[tex]\[ 3(x+7)^2 - 32 + 32 = -29 + 32 \][/tex]
This simplifies to:
[tex]\[ 3(x+7)^2 = 3 \][/tex]
2. Divide both sides by 3:
[tex]\[ \frac{3(x+7)^2}{3} = \frac{3}{3} \][/tex]
This simplifies to:
[tex]\[ (x+7)^2 = 1 \][/tex]
3. Solve the resulting equation:
Take the square root of both sides:
[tex]\[ x+7 = \pm 1 \][/tex]
This gives us two equations:
[tex]\[ x+7 = 1 \quad \text{or} \quad x+7 = -1 \][/tex]
4. Solve for [tex]\( x \)[/tex] in both cases:
- For [tex]\( x+7 = 1 \)[/tex]:
[tex]\[ x = 1 - 7 \][/tex]
[tex]\[ x = -6 \][/tex]
- For [tex]\( x+7 = -1 \)[/tex]:
[tex]\[ x = -1 - 7 \][/tex]
[tex]\[ x = -8 \][/tex]
Thus, the solutions to the equation [tex]\( 3(x+7)^2 - 32 = -29 \)[/tex] are:
[tex]\[ x = -8 \quad \text{and} \quad x = -6 \][/tex]
1. Simplify the equation:
[tex]\[ 3(x+7)^2 - 32 = -29 \][/tex]
Add 32 to both sides to isolate the quadratic term:
[tex]\[ 3(x+7)^2 - 32 + 32 = -29 + 32 \][/tex]
This simplifies to:
[tex]\[ 3(x+7)^2 = 3 \][/tex]
2. Divide both sides by 3:
[tex]\[ \frac{3(x+7)^2}{3} = \frac{3}{3} \][/tex]
This simplifies to:
[tex]\[ (x+7)^2 = 1 \][/tex]
3. Solve the resulting equation:
Take the square root of both sides:
[tex]\[ x+7 = \pm 1 \][/tex]
This gives us two equations:
[tex]\[ x+7 = 1 \quad \text{or} \quad x+7 = -1 \][/tex]
4. Solve for [tex]\( x \)[/tex] in both cases:
- For [tex]\( x+7 = 1 \)[/tex]:
[tex]\[ x = 1 - 7 \][/tex]
[tex]\[ x = -6 \][/tex]
- For [tex]\( x+7 = -1 \)[/tex]:
[tex]\[ x = -1 - 7 \][/tex]
[tex]\[ x = -8 \][/tex]
Thus, the solutions to the equation [tex]\( 3(x+7)^2 - 32 = -29 \)[/tex] are:
[tex]\[ x = -8 \quad \text{and} \quad x = -6 \][/tex]
