Express 3x squared + 2x + 1 in the form a(x+p)squared + q where a,p and q are real numbers
3 x^2 + 2 x + 1
3 [x^2 + 2/3 x + 1/3]
3 [ x^2 + 2/3 x + 4/36 + 1/3 - 4/36 ]
3 [ (x+2/6)^2 + 3/9 - 1/9 ]
3 [ (x+1/3)^2 + 2/9 ]
3 (x+1/3)^2 + 2/3
I used "completing the square" which you will find in your text under solving quadratic equations.
To express the quadratic expression 3x^2 + 2x + 1 in the form a(x + p)^2 + q, we need to complete the square. Here's how to do it:
Step 1: Take the coefficient of x^2 (in this case, 3) and factor it out from the quadratic expression:
3x^2 + 2x + 1 = 3(x^2 + (2/3)x) + 1
Step 2: To complete the square, we need to add and subtract the square of half the coefficient of x inside the parentheses. In this case, half of (2/3) is (1/3).
Adding and subtracting (1/3)^2 = 1/9 inside the parentheses:
3(x^2 + (2/3)x + 1/9 - 1/9) + 1
Step 3: We can now rewrite the expression inside the parentheses as a perfect square trinomial:
3(x + 1/3)^2 - 1/3 + 1
Step 4: Simplify the constant terms:
3(x + 1/3)^2 - 1/3 + 3/3
Step 5: Combine the constant terms:
3(x + 1/3)^2 + 2/3
So, the quadratic expression 3x^2 + 2x + 1 can be written in the desired form as a(x + p)^2 + q, where a = 3, p = 1/3, and q = 2/3.