Q. What is the solution of the equation \(x^2 = 64\)?

Problem

Solve the equation

\(x^{2} = 64\)

Step-by-step solution

1. Recognize the equation. We are given the quadratic equation

   \(x^{2} = 64\).

2. Bring all terms to one side (optional, useful for factoring):

   \(x^{2} – 64 = 0\).

   This expresses the equation in the standard form “something equals zero”, which allows using the zero-product property after factoring.

3. Factor the difference of squares. Use the identity \(a^{2}-b^{2}=(a-b)(a+b)\). With \(a=x\) and \(b=8\) we get

   \(x^{2}-64 = (x-8)(x+8)\).

4. Apply the zero-product property: if a product of two real numbers is zero, then at least one factor is zero. Thus

   \((x-8)(x+8)=0\)

   implies either

   \(x-8=0\)

   or

   \(x+8=0\).

5. Solve each linear equation:

   If \(x-8=0\) then \(x=8\).

   If \(x+8=0\) then \(x=-8\).

6. Alternatively, use the square-root property: for a nonnegative number \(a\), the solutions of \(x^{2}=a\) are \(x=\pm\sqrt{a}\). Applying this with \(a=64\) gives

   \(x=\pm\sqrt{64}=\pm 8\).

7. Check both solutions by substitution:

   For \(x=8\): \(8^{2}=64\), so it satisfies the equation.

   For \(x=-8\): \((-8)^{2}=64\), so it also satisfies the equation.

8. Conclusion: the solutions are

   \(\boxed{x=8}\) and \(\boxed{x=-8}\).

   As a solution set: \(\{ -8,\, 8 \}\).