Sure, let's solve the equation [tex]\(4(x^2 - 3)^3 = 1372\)[/tex] step-by-step.
### Step 1: Simplify the equation
First, we'll start by isolating the term involving [tex]\(x\)[/tex].
[tex]\[4(x^2 - 3)^3 = 1372\][/tex]
Divide both sides by 4:
[tex]\[(x^2 - 3)^3 = \frac{1372}{4}\][/tex]
[tex]\[(x^2 - 3)^3 = 343\][/tex]
### Step 2: Take the cube root of both sides
Next, we need to eliminate the cube by taking the cube root of both sides of the equation:
[tex]\[x^2 - 3 = \sqrt[3]{343}\][/tex]
[tex]\[x^2 - 3 = 7\][/tex]
### Step 3: Solve for [tex]\(x^2\)[/tex]
Add 3 to both sides to isolate [tex]\(x^2\)[/tex]:
[tex]\[x^2 = 7 + 3\][/tex]
[tex]\[x^2 = 10\][/tex]
### Step 4: Solve for [tex]\(x\)[/tex]
To find [tex]\(x\)[/tex], we need to take the square root of both sides. This gives us two solutions because both positive and negative roots satisfy the equation:
[tex]\[x = \pm\sqrt{10}\][/tex]
Thus, the real solutions are:
[tex]\[x = \sqrt{10} \quad \text{and} \quad x = -\sqrt{10}\][/tex]
### Step 5: Consider the complex roots
The equation under consideration is a third-degree polynomial equation, which means there might be complex roots as well. For completeness, we need to use the theory that complex roots come in conjugate pairs. Solving this as a polynomial could generate complex roots.
From solving [tex]\(4(x^2 - 3)^3 = 1372\)[/tex], we have:
[tex]\[ x^2 - 3 = 7 \][/tex]
[tex]\[ x^2 = 10 \][/tex]
We can also solve the more generalized form of the polynomial for the root pairs:
In terms of complex roots, solving complex cubic equations can typically yield results involving imaginary numbers.
When simplified, the solutions include complex roots which aren't necessarily straightforward from typical simplifications but result in the complete set:
[tex]\[ x = -\sqrt{-\frac{1}{2} - \frac{7\sqrt{3}i}{2}}, \ x = \sqrt{-\frac{1}{2} - \frac{7\sqrt{3}i}{2}} \][/tex]
and
[tex]\[ x = -\sqrt{-\frac{1}{2} + \frac{7\sqrt{3}i}{2}}, \ x = \sqrt{-\frac{1}{2} + \frac{7\sqrt{3}i}{2}} \][/tex]
So, in summary, the complete set of solutions for the original equation [tex]\(4(x^2 - 3)^3 = 1372\)[/tex] is:
[tex]\[x = \sqrt{10}, \, -\sqrt{10}\][/tex]
[tex]\[x = \sqrt{-\frac{1}{2} - \frac{7\sqrt{3}i}{2}}, \, -\sqrt{-\frac{1}{2} - \frac{7\sqrt{3}i}{2}}\][/tex]
[tex]\[x = \sqrt{-\frac{1}{2} + \frac{7\sqrt{3}i}{2}}, \, -\sqrt{-\frac{1}{2} + \frac{7\sqrt{3}i}{2}}\][/tex]
These cover all possible real and complex solutions to the given equation.
