Q. \(2(x^2-7)+3=-3\)

Step-by-step solution

1. Start with the given equation:
   \[ 2(x^2 – 7) + 3 = -3 \]
2. Eliminate the constant +3 on the left by subtracting 3 from both sides. This keeps the equality true because the same operation is applied to both sides.
   \[ 2(x^2 – 7) + 3 – 3 = -3 – 3 \]
   Simplify each side:
   \[ 2(x^2 – 7) = -6 \]
3. Remove the coefficient 2 multiplying the parentheses by dividing both sides by 2. Division by a nonzero number preserves equality.
   \[ \frac{2(x^2 – 7)}{2} = \frac{-6}{2} \]
   Simplify:
   \[ x^2 – 7 = -3 \]
4. Isolate \(x^2\) by adding 7 to both sides:
   \[ x^2 – 7 + 7 = -3 + 7 \]
   Simplify:
   \[ x^2 = 4 \]
5. Take the square root of both sides. When taking square roots in an equation, include both the positive and negative roots because both squared give the same value.
   \[ x = \pm\sqrt{4} \]
   Simplify the square root:
   \[ x = \pm 2 \]
6. List the solutions: \(x = 2\) and \(x = -2\)
7. Optional verification: substitute each solution into the original equation to confirm they satisfy it.
   For \(x = 2\):
   \[ 2(2^2 – 7) + 3 = 2(4 – 7) + 3 = 2(-3) + 3 = -6 + 3 = -3 \quad \checkmark \]
   For \(x = -2\):
   \[ 2((-2)^2 – 7) + 3 = 2(4 – 7) + 3 = -6 + 3 = -3 \quad \checkmark \]

Final answer: \(x = 2\) or \(x = -2\) (solution set: \(\{-2, 2\}\))