it is called completing the square

here is the equation 4x^2-4x+1=9 if anyone can help me please do okay i really need ot understand this

4x^2-4x+1=9
Move the numbers to the other side.
4x^2 -4x = 9-1
4x^2 -4x = 8
Now factor out 4 to make things easier.
x^2 -x =2 but remember we have all of that multiplied by 4.
The completed square goes where I place the Y.
x^2 - x +Y = 2 and to keep everything equal we must add Y to the right side, also.
x^2 - x + Y = 2 + Y
To determine what number completes the square, take 1/2 the x term and square it. 1/2 of 1 is 1/2 and 1/2 squared is 1/4. Now add 1/4 to both sides like this.
x^2 - x + 1/4 = 2 + 1/4 = 2.25
Now factor the left side. It is a perfect square now so it will be
(x - 1/2)^2 = 2.25
Now take the square root of both sides and solve for x. I hope this helps.

solving polynomial long division

20

To solve a polynomial long division problem, follow these steps:

1. Write the dividend (the polynomial being divided) and the divisor (the polynomial dividing the dividend) in standard form, with the terms arranged in descending order of their degrees.
2. Divide the first term of the dividend by the first term of the divisor to determine the first term of the quotient.
3. Multiply the entire divisor by the first term of the quotient and subtract the result from the dividend.
4. Bring down the next term from the dividend and repeat steps 2 and 3 until all terms have been divided.
5. The resulting polynomial after the long division is the quotient and any remaining terms are the remainder.

Here's an example to illustrate the process:

Divide 2x^3 + 3x^2 - 5x + 1 by x - 2.

Step 1: Write the dividend and divisor in standard form.
Dividend: 2x^3 + 3x^2 - 5x + 1
Divisor: x - 2

Step 2: Divide the first terms of the dividend and divisor.
2x^3 / x = 2x^2 (first term of the quotient)

Step 3: Multiply the entire divisor by the first term of the quotient and subtract from the dividend.
Multiply: (x - 2) * 2x^2 = 2x^3 - 4x^2
Subtract: (2x^3 + 3x^2 - 5x + 1) - (2x^3 - 4x^2) = 7x^2 - 5x + 1

Step 4: Bring down the next term and repeat steps 2 and 3.
7x^2 / x = 7x (second term of the quotient)
Multiply: (x - 2) * 7x = 7x^2 - 14x
Subtract: (7x^2 - 5x + 1) - (7x^2 - 14x) = 9x + 1

Step 5: Bring down the next term and repeat steps 2 and 3.
9x / x = 9 (third term of the quotient)
Multiply: (x - 2) * 9 = 9x - 18
Subtract: (9x + 1) - (9x - 18) = 19

Since there are no more terms to bring down, the division is complete. The resulting quotient is 2x^2 + 7x + 9 and the remainder is 19.