Solve the equation 3x^2 - 48 = 0.

Step-by-step solution

1. 1

   Notice there is no $x$ term, so the quadratic formula is unnecessary. An equation of the form $ax^{2}+c=0$ can be solved by isolating $x^{2}$ and taking roots — much faster than substituting into the formula. Start by adding $48$ to both sides:

   $3x^{2}=48$

2. 2

   Divide by the leading coefficient. Divide both sides by $3$ before taking any root; taking the root of $3x^{2}$ directly would leave an awkward $\sqrt{3}$:

   $x^{2}=16$

3. 3

   Apply the square root property with both signs. If $x^{2}=k$ with $k>0$, then $x=\pm\sqrt{k}$ — two solutions, because squaring destroys sign information:

   $x=\pm\sqrt{16}=\pm4$

   Writing only $x=4$ loses half the answer. The symbol $\sqrt{16}$ alone means the principal (non-negative) root $4$, which is why the $\pm$ has to be inserted by hand.

4. 4

   Verify both roots in the original equation. At $x=4$: $3(16)-48=48-48=0$ ✓. At $x=-4$: $3(-4)^{2}-48=3(16)-48=0$ ✓. Note $(-4)^{2}=+16$ — the square of a negative is positive, which is precisely why the negative root works too.

5. 5

   Cross-check by factoring. $3x^{2}-48=3(x^{2}-16)=3(x-4)(x+4)$, and a product is zero only when a factor is zero, giving $x=4$ or $x=-4$ ✓. Vieta agrees as well: the roots sum to $0=-b/a$ (there is no $x$ term) and multiply to $-48/3=-16$ ✓.