Factorise ab^2(x^2+2)=2x(b^3)+3a^3 using quadratic formula

To factorize the expression \( ab^2(x^2 + 2) = 2x(b^3) + 3a^3 \) using the quadratic formula, we need to express it in a form where applying the quadratic formula makes sense.

### Step 1: Rewrite the Expression in Quadratic Form

First, let's rewrite the equation in a standard quadratic form:

\[
ab^2(x^2 + 2) - 2xb^3 - 3a^3 = 0
\]

Expand and simplify to arrange terms by powers of \(x\):

\[
ab^2x^2 + 2ab^2 - 2xb^3 - 3a^3 = 0
\]

This is now a quadratic equation in terms of \(x\). The general form of a quadratic equation is:

\[
Ax^2 + Bx + C = 0
\]

where:

\[
A = ab^2, \quad B = -2b^3, \quad C = 2ab^2 - 3a^3
\]

### Step 2: Apply the Quadratic Formula

The quadratic formula for finding the roots \(x\) of the equation \(Ax^2 + Bx + C = 0\) is:

\[
x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}
\]

Substitute \(A = ab^2\), \(B = -2b^3\), and \(C = 2ab^2 - 3a^3\) into the quadratic formula:

\[
x = \frac{-(-2b^3) \pm \sqrt{(-2b^3)^2 - 4(ab^2)(2ab^2 - 3a^3)}}{2(ab^2)}
\]

Simplify this step-by-step:

\[
x = \frac{2b^3 \pm \sqrt{(4b^6) - 4ab^2(2ab^2 - 3a^3)}}{2ab^2}
\]

Simplify inside the square root:

\[
x = \frac{2b^3 \pm \sqrt{4b^6 - 4ab^2(2ab^2) + 4ab^2(3a^3)}}{2ab^2}
\]

\[
x = \frac{2b^3 \pm \sqrt{4b^6 - 8a(b^4) + 12a^4b^2}}{2ab^2}
\]

### Step 3: Factor the Expression

To factorize, observe the discriminant (\(\Delta\)) under the square root and simplify where possible:

\[
x = \frac{2b^3 \pm \sqrt{4b^6 - 8ab^4 + 12a^4b^2}}{2ab^2}
\]

If the discriminant is a perfect square, we can fully factorize. Here, it needs a form like:

\[
\sqrt{(2b^3 - 2ab^2 + 2a^2)^2}
\]

or equivalent, if it represents perfect factorization. The factors here directly aren't simple given the structure unless particular roots or forms simplify further:

### Step 4: Write the Factored Form

Thus, roots, \(x_1\) and \(x_2\):

\[
x_1 = \frac{2b^3 + \sqrt{4b^6 - 8ab^4 + 12a^4b^2}}{2ab^2}
\]

\[
x_2 = \frac{2b^3 - \sqrt{4b^6 - 8ab^4 + 12a^4b^2}}{2ab^2}
\]

And factorized form:

\[
ab^2 \left( x - x_1 \right) \left( x - x_2 \right)
\]

**Would you like a deeper breakdown or further clarification on these steps?**

### Related Questions to Explore

1. How can we simplify the square root terms in discriminants involving multiple parameters?
2. Are the results still valid under different conditions, such as assuming integer values?
3. What other methods work for factorizing polynomials with cross-product terms or mixed variables?
4. Can specific values for \(a\) or \(b\) result in simpler factors or roots?
5. What role does completing the square play in polynomial expressions?

**Tip:** Always ensure the quadratic form aligns; sometimes adjustments or factoring simplify these expressions.

Learn how to factorize the equation ab^2(x^2 + 2) = 2x(b^3) + 3a^3 using the quadratic formula. This problem covers advanced algebraic concepts, including quadratic equations, factoring, and discriminants. Suitable for students in Grades 10-12.

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