Q. \(2(x^2 – 6) – 8 = 2\)

Answer

Solve: \(2(x^2-6)-8=2\).

Stepwise:
\[
2x^2-12-8=2 \implies 2x^2-20=2 \implies 2x^2=22 \implies x^2=11.
\]

So \(x=\pm\sqrt{11}\) (approximately \(x\approx\pm3.317\)).

Detailed Explanation

Solution

Write the equation:
\[2(x^2-6)-8=2\]
Distribute the 2 into the parentheses (multiply each term by 2):
\[2x^2-12-8=2\]

Explanation: \(2\cdot x^2=2x^2\) and \(2\cdot(-6)=-12\).
Combine like terms on the left:
\[2x^2-20=2\]

Explanation: \(-12-8=-20\).
Isolate the quadratic term by adding 20 to both sides:
\[2x^2=22\]
Divide both sides by 2:
\[x^2=11\]
Take the square root of both sides, remembering both positive and negative roots:
\[x=\pm\sqrt{11}\]
Check (quick verification):
\[2\big((\pm\sqrt{11})^2-6\big)-8=2(11-6)-8=2\cdot5-8=10-8=2,\]

so both \(x=\sqrt{11}\) and \(x=-\sqrt{11}\) satisfy the equation.

Final answer: \(x=\pm\sqrt{11}\)

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FAQs

Q How do I simplify \(2(x^2-6)-8=2\) step by step?

A Distribute: \(2x^2-12-8=2\). Combine like terms: \(2x^2-20=2\). Add 20: \(2x^2=22\). Divide by 2: \(x^2=11\). Take square roots: \(x=\pm\sqrt{11}\).

Q How do I correctly distribute the 2 in \(2(x^2-6)\)?

A Multiply each term inside: \(2(x^2)=2x^2\) and \(2(-6)=-12\). So \(2(x^2-6)=2x^2-12\).

Q How do I combine the constants \(-12\) and \(-8\)?

A Add them: \(-12-8=-20\). So the left side becomes \(2x^2-20\).

Q How do I isolate \(x^2\) from \(2x^2-20=2\)?

A Add 20 to both sides to get \(2x^2=22\). Then divide by 2: \(x^2=11\).

Q Why do I get two solutions \(x=\pm\sqrt{11}\)?

A Because \(x^2=11\) means both a positive and a negative number squared give 11. Taking square roots yields two real solutions: \(x=\sqrt{11}\) and \(x=-\sqrt{11}\).

Q Can I factor the equation instead of taking square roots?

A Yes. Rearrange to \(x^2-11=0\). Factor as \((x-\sqrt{11})(x+\sqrt{11})=0\). Set each factor zero to get \(x=\pm\sqrt{11}\).

Q Do I need the quadratic formula here?

A Not really: the equation is already in the simple form \(x^2=11\). The quadratic formula would give the same result, \(x=\pm\sqrt{11}\), but is unnecessary for this easy case.

Q How do I check my solutions quickly?

A Substitute \(x=\pm\sqrt{11}\) into the original: \(2((\pm\sqrt{11})^2-6)-8=2(11-6)-8=2\cdot5-8=10-8=2\). Both satisfy the equation.

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