Question

question
solve for c. express your answer as an integer or integers or in simplest radical form.
\\(-88 = -6c^3 - 4\\)
answer attempt 1 out of 2
+ additional solution - no solution
c = |
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Explanation:

Step1: Add 4 to both sides

To isolate the term with \( c^3 \), we add 4 to both sides of the equation \(-88 = -6c^3 - 4\). This gives us \(-88 + 4 = -6c^3\), which simplifies to \(-84 = -6c^3\).

Step2: Divide both sides by -6

Next, we divide both sides of the equation \(-84 = -6c^3\) by -6. So, \(\frac{-84}{-6} = c^3\), which simplifies to \(14 = c^3\)? Wait, no, wait, -88 +4 is -84, and -84 divided by -6 is 14? Wait, no, wait, let's check again. Wait, the original equation is \(-88 = -6c^3 -4\). So adding 4 to both sides: \(-88 + 4 = -6c^3\), so \(-84 = -6c^3\). Then divide both sides by -6: \(\frac{-84}{-6} = c^3\), so \(14 = c^3\)? Wait, that can't be right. Wait, maybe I made a mistake. Wait, no, wait, let's re-express the equation. Wait, the equation is \(-88 = -6c^3 - 4\). Let's add 4 to both sides: \(-88 + 4 = -6c^3\), so \(-84 = -6c^3\). Then divide both sides by -6: \(c^3 = \frac{-84}{-6} = 14\)? Wait, but 14 is not a perfect cube. Wait, maybe I misread the equation. Wait, the original equation is \(-88 = -6c^3 - 4\)? Let me check again. Oh! Wait, maybe it's \(-88 = -6c^2 -4\)? No, the user wrote \(c^3\). Wait, maybe the equation is \(-88 = -6c^3 - 4\). Let's solve it again.

Wait, let's do it step by step:

1. Start with \(-88 = -6c^3 - 4\)
2. Add 4 to both sides: \(-88 + 4 = -6c^3\) → \(-84 = -6c^3\)
3. Divide both sides by -6: \(c^3 = \frac{-84}{-6} = 14\)
4. Then \(c = \sqrt[3]{14}\). But that's not an integer. Wait, maybe the equation was supposed to be \(-88 = -6c^2 -4\)? Let's check. If it's \(c^2\), then:

1. \(-88 = -6c^2 -4\)
2. Add 4: \(-84 = -6c^2\)
3. Divide by -6: \(14 = c^2\)
4. Then \(c = \pm\sqrt{14}\), but that's not an integer either. Wait, maybe the original equation is \(-88 = -6c^3 - 4\), but maybe there's a typo. Wait, maybe the equation is \(-88 = -6c^3 + 4\)? No, the user wrote \(-6c^3 -4\). Wait, maybe I made a mistake in the sign. Let's check again.

Wait, the equation is \(-88 = -6c^3 - 4\). Let's add 4 to both sides: \(-88 + 4 = -6c^3\) → \(-84 = -6c^3\). Then divide both sides by -6: \(c^3 = 14\). So \(c = \sqrt[3]{14}\), but that's not an integer. Wait, maybe the equation is \(-88 = -6c^3 - 4\) with a different number. Wait, maybe the original equation is \(-88 = -6c^3 - 4\), but maybe the user made a typo. Alternatively, maybe I misread the equation. Wait, let's check the original problem again. The user wrote: \(-88 = -6c^3 - 4\). So solving for \(c\):

\(-88 + 4 = -6c^3\)

\(-84 = -6c^3\)

\(c^3 = 14\)

\(c = \sqrt[3]{14}\)

But that's not an integer. Wait, maybe the equation is \(-88 = -6c^3 + 4\)? Let's try that.

If it's \(-88 = -6c^3 + 4\), then:

Subtract 4: \(-92 = -6c^3\)

Divide by -6: \(c^3 = \frac{92}{6} = \frac{46}{3}\), still not integer.

Wait, maybe the equation is \(-88 = -6c^3 - 4\), but the intended equation was \(-88 = -6c^3 - 4\) with a different constant. Wait, maybe the original equation is \(-88 = -6c^3 - 4\), but let's check the arithmetic again.

Wait, -88 + 4 is -84, divided by -6 is 14. So \(c^3 = 14\), so \(c = \sqrt[3]{14}\). But the problem says "Express your answer as an integer or integers or in simplest radical form." So \(\sqrt[3]{14}\) is in simplest radical form. But maybe the equation was supposed to be \(-88 = -6c^3 - 4\) with a different number. Wait, maybe the equation is \(-88 = -6c^3 - 4\), but perhaps I made a mistake. Wait, let's check with \(c = 2\): \(-6*(2)^3 -4 = -6*8 -4 = -48 -4 = -52\), not -88. \(c = 3\): \(-6*27 -4 = -162 -4 = -166\), not -88. \(c = \sqrt[3]{14}\) is approximately 2.41. So maybe the equation is correct, and the answer is \(\sqrt[3]{14}\). But that seems odd. Wait, maybe the original equation is \(-88 = -6c^2 -4\)? Let's try that.

If it's \(c^2\):

\(-88 = -6c^2 -4\)

Add 4: \(-84 = -6c^2\)

Divide by -6: \(c^2 = 14\)

\(c = \pm\sqrt{14}\)

But that's also not an integer. Wait, maybe the equation is \(-88 = -6c^3 - 4\) with a different constant. Wait, maybe the user made a typo. Alternatively, maybe I misread the equation. Wait, the original equation is \(-88 = -6c^3 - 4\). Let's solve it again:

1. Add 4 to both sides: \(-88 + 4 = -6c^3\) → \(-84 = -6c^3\)
2. Divide both sides by -6: \(c^3 = 14\)
3. Take the cube root: \(c = \sqrt[3]{14}\)

But the problem says "Express your answer as an integer or integers or in simplest radical form." So \(\sqrt[3]{14}\) is the answer. But maybe the equation was supposed to be \(-88 = -6c^3 - 4\) with a different number. Wait, maybe the original equation is \(-88 = -6c^3 - 4\), and the answer is \(\sqrt[3]{14}\). But that's not an integer. Wait, maybe I made a mistake in the sign. Let's check again.

Wait, \(-88 = -6c^3 - 4\)

Add 4: \(-84 = -6c^3\)

Divide by -6: \(c^3 = 14\)

Yes, that's correct. So \(c = \sqrt[3]{14}\). But maybe the equation was \(-88 = -6c^3 + 4\)? Let's try that:

\(-88 = -6c^3 + 4\)

Subtract 4: \(-92 = -6c^3\)

Divide by -6: \(c^3 = \frac{92}{6} = \frac{46}{3}\), still not integer.

Alternatively, maybe the equation is \(-88 = -6c^3 - 4\) with a different coefficient. Wait, maybe the equation is \(-88 = -6c^3 - 4\), and the answer is \(\sqrt[3]{14}\). So I think that's the solution.

Wait, but the user might have intended a different equation. For example, if the equation was \(-88 = -6c^3 + 4\), then:

\(-88 - 4 = -6c^3\)

\(-92 = -6c^3\)

\(c^3 = \frac{92}{6} = \frac{46}{3}\), still not integer.

Alternatively, if the equation was \(-88 = -6c^3 - 4\) with \(c^2\), then:

\(-88 = -6c^2 - 4\)

\(-84 = -6c^2\)

\(c^2 = 14\)

\(c = \pm\sqrt{14}\)

But that's not an integer either. So maybe the original equation is correct, and the answer is \(\sqrt[3]{14}\).

Answer:

\(c = \sqrt[3]{14}\)

Wait, but maybe I made a mistake. Let me check again. Wait, the original equation is \(-88 = -6c^3 - 4\). Let's plug \(c = \sqrt[3]{14}\) back into the equation:

Left side: -88

Right side: -6(\sqrt[3]{14})^3 - 4 = -614 - 4 = -84 - 4 = -88. Yes, that works. So the solution is \(c = \sqrt[3]{14}\).