Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ Solve algebraically for all values of [tex]x[/tex]:

[tex]x^5 - 19x^4 + 90x^3 = 0[/tex]

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].