Answer :

To solve the equation [tex]\(x^5 - 19x^4 + 90x^3 = 0\)[/tex] algebraically, we can follow these steps:

1. Factor out the common term:

The equation [tex]\(x^5 - 19x^4 + 90x^3 = 0\)[/tex] has a common factor of [tex]\(x^3\)[/tex] in each term. We can factor this out:

[tex]\[
x^3(x^2 - 19x + 90) = 0
\][/tex]

2. Solve the factored equation:

For the factored equation [tex]\(x^3(x^2 - 19x + 90) = 0\)[/tex], we have a product of factors that equals zero. According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero.

- Solve [tex]\(x^3 = 0\)[/tex]:

This factor simplifies to [tex]\(x = 0\)[/tex]. So, one of the solutions is [tex]\(x = 0\)[/tex].

- Solve the quadratic equation [tex]\(x^2 - 19x + 90 = 0\)[/tex]:

We can use the quadratic formula to find the roots of the quadratic equation. The quadratic formula is:

[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]

Here, [tex]\(a = 1\)[/tex], [tex]\(b = -19\)[/tex], and [tex]\(c = 90\)[/tex]. Plug these values into the formula:

[tex]\[
x = \frac{-(-19) \pm \sqrt{(-19)^2 - 4 \times 1 \times 90}}{2 \times 1}
\][/tex]

[tex]\[
x = \frac{19 \pm \sqrt{361 - 360}}{2}
\][/tex]

[tex]\[
x = \frac{19 \pm \sqrt{1}}{2}
\][/tex]

[tex]\[
x = \frac{19 \pm 1}{2}
\][/tex]

This gives us two solutions:

[tex]\[
x = \frac{19 + 1}{2} = 10
\][/tex]

[tex]\[
x = \frac{19 - 1}{2} = 9
\][/tex]

3. List all solutions:

Thus, the solutions to the equation [tex]\(x^5 - 19x^4 + 90x^3 = 0\)[/tex] are [tex]\(x = 0\)[/tex], [tex]\(x = 9\)[/tex], and [tex]\(x = 10\)[/tex].

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Rewritten by : Barada