Question

100 POINTS
Solve x4 - 10x² + 9 = 0.

Answer 1

`x=\pm1 \:\textsf{or}\:x=\pm3`

Explanation:

`x^4-10x^2+9=0`

`\textsf{let}\:x^2=y`

`\implies y^2-10y+9=0`

To factor a quadratic in the form
`ax^2+bx+c`, find 2 two numbers that multiply to
`ac` and sum to
`b`.

⇒ two numbers that multiply to 9 and sum to -10: -9 and -1

Rewrite
`b` as the sum of these 2 numbers:

`\implies y^2-y-9y+9=0`

Factorize the first two terms and the last two terms separately:

`\implies y(y-1)-9(y-1)=0`

Factor out the common term
`(y-1)`:

`\implies (y-9)(y-1)=0`

`\implies (y-9)=0 \implies y=9`

`\implies (y-1)=0 \implies y=1`

Change
`y` back to
`x^2`:

`\implies x^2=9\implies x=\pm3`

`\implies x^2=1\implies x=\pm 1`

`\textsf{So}\: x=\pm1 \:\textsf{or}\:x=\pm3`

Answer 2

`$x=\pm1\;\textrm{or}\;x=\pm3$`

Explanation:

Substitute
`u = x^2` into the equation. This will make the quadratic formula easy to use.

`\implies u^2 - 10u + 9 = 0`

Consider the form
`x^2 + bx + c = 0`. Find a pair of integers whose product is
`c` and whose sum is
`c`. In this case, whose product is
`9`and whose sum is
`10`. And write the factored form using the integers.

`\implies (u - 9)(u - 1) = 0`

If any individual factor on the left side of the equation is equal to
`0`, the entire expression will be equal to
`0`.

`\implies \begin{cases}u - 9 = 0 \\ \\ u - 1 = 0\end{cases}`

Set
`u - 9` equal to
`0` and solve for
`u`. Similarly, Set
`u - 1` equal to
`0` and solve for
`u`.

`\implies \begin{cases}u = 9 \\ \\ u = 1\end{cases}`

Substitute the real value of
`u = x^2` back into the solved equation.

`\implies \begin{cases}x^2 = 9 \\ \\ x^2 = 1\end{cases}`

Take the square root of both sides of the equations to eliminate the exponent on the left side.

`\implies \begin{cases}x = √(9) \\ \\ x = √(1)\end{cases}`

Solve both the equations for the value of
`x`.

`\implies \begin{cases}x = \pm3 \\ \\ x = \pm1\end{cases}`

Hence, this is our required solution for this question.

`\rule{300}{1}`

Additional information:

There are three ways to solve a quadratic equation:

1. Factoring
2. Complete the Square
3. Quadratic Formula

Factoring uses the logic that the product of any number and zero is zero. Complete the square method uses square root, and the quadratic formula is the simpler method of it.

Quadratic Formula
`x=(-b\pm√(b^2-4ac) )/(2a)`