how do i put x^4 + 5x^2 -8 into quadratic form?
thnax soooo much!
To put the expression x^4 + 5x^2 - 8 into quadratic form, we need to manipulate it so that it resembles a quadratic equation of the form ax^2 + bx + c. Here's how you can do it step by step:
1. Start by identifying the highest power of x in the expression, which in this case is x^4.
2. Let's introduce a new variable to simplify things. Set y = x^2.
3. Rewrite the original expression in terms of y: y^2 + 5y - 8.
4. Now, we have a quadratic equation in terms of y. To solve for y, we can use factoring, completing the square, or the quadratic formula.
a. Factoring: Unfortunately, the quadratic expression y^2 + 5y - 8 cannot be factored easily, so this method doesn't work.
b. Completing the square: To complete the square, we need to add and subtract the square of half the coefficient of the y-term, which is (5/2)^2 = 25/4.
y^2 + 5y - 8 + 25/4 - 25/4
(y + 5/2)^2 - 8 + 25/4 - 25/4
(y + 5/2)^2 - 32/4
(y + 5/2)^2 - 8
5. Now, we can rewrite the equation back in terms of x, replacing y with x^2:
(x^2 + 5/2)^2 - 8
Therefore, the expression x^4 + 5x^2 - 8 can be put in quadratic form as (x^2 + 5/2)^2 - 8.
let y = x^2
then
y^2 + 5 y - 8