Q. What are the roots of \( f(x) = x^2 – 48 \)?

Answer

Isolate the squared term.
Add 48 to both sides.

\[ x^2 = 48 \]
Take the square root.
Remember to include both positive and negative roots.

\[ x = \pm\sqrt{48} \]
Simplify the radical.
Factor out the perfect square 16.

\[ x = \pm 4\sqrt{3} \]

Detailed Explanation

Solution — step by step

Write the equation for the roots by setting the polynomial equal to zero: \(x^2 – 48 = 0\).
Isolate the squared term by adding 48 to both sides: \(x^2 = 48\).
Take the square root of both sides. Because the square root operation yields two possible signs (positive and negative), we obtain
\[
x = \pm \sqrt{48}.
\]
Simplify the radical. Factor 48 as \(48 = 16 \cdot 3\), so
\[
\sqrt{48} = \sqrt{16\cdot 3} = \sqrt{16}\,\sqrt{3} = 4\sqrt{3}.
\]
Therefore
\[
x = \pm 4\sqrt{3}.
\]
Verify by substitution (optional check). For \(x = 4\sqrt{3}\):
\[
f(x) = (4\sqrt{3})^2 – 48 = 16\cdot 3 – 48 = 48 – 48 = 0.
\]
For \(x = -4\sqrt{3}\):
\[
f(x) = (-4\sqrt{3})^2 – 48 = 16\cdot 3 – 48 = 0.
\]
Both values satisfy the equation, so they are the roots.

Final answer: \(x = 4\sqrt{3}\) and \(x = -4\sqrt{3}\).

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Frequently Asked Questions

What are the roots of f(x) = x^2 - 48?

Solve x^2 - 48 = 0 => x^2 = 48, so x = ±sqrt(48) = ±4sqrt3.

How do you solve it using the square-root method?

Set x^2 = 48 and take square roots: x = ±sqrt(48). Simplify sqrt(48) = 4sqrt3. Always include both the positive and negative roots.

Can the expression be factored?

Yes: x^2 - 48 = (x - 4sqrt3)(x + 4sqrt3), factoring over the real numbers using the found roots.

Are the roots rational, irrational, or complex?

They are irrational real numbers because sqrt(48) simplifies to 4sqrt3 and sqrt3 is irrational. No complex roots beyond these two real ones.

What does the discriminant tell us?

Discriminant = b^2 - 4ac = 0^2 - 4(1)(-48) = 192 > 0, so there are two distinct real roots.

What are the decimal approximations of the roots?

sqrt(48) ≈ 6.9282, so the roots are approximately x ≈ 6.9282 and x ≈ -6.9282.

How many times does each root occur (multiplicity)?

Each root has multiplicity 1 because the factorization yields two distinct linear factors.

What is the graph shape, axis of symmetry, and vertex?

The graph is an upward-opening parabola. Axis of symmetry is x = 0. Vertex at (0, -48), the minimum point of the parabola.

How can I verify root quickly?

Substitute x = 4sqrt3 (or -4sqrt3) into x^2 - 48: (4sqrt3)^2 - 48 = 48 - 48 = 0, confirming it's root.

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