Write the function f(x) = ax^2 + bx + c in completed square form. SHOW YOUR WORK!!!
How about you showing the first two steps, and I will continue from there ?
i dont know how
Now is 0:37 AM in Bosnia.
I have to go to sleep and I'll write a solution for you.
f(x) = a x² + b x + c =
a [ x² + ( b / a ) x ] + c =
a [ x² + 2 ( b / 2a ) x ] + c + ( b / 2a )² - ( b / 2a )² =
a [ x² + 2 ( b / 2a ) x + ( b / 2a )² ] + c - ( b / 2a )² =
a ( x + b / 2a )² + c - ( b / 2a )²
f(x) = a x² + b x + c = a ( x + b / 2a )² + c - ( b / 2a )²
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Remark:
( m + n )² = m² + 2 ∙ m ∙ n + n²
In tis case:
( x + b / 2a )² = x² + 2 ∙ ( b / 2a ) ∙ x + ( b / 2a )²
So
x² + 2 ( b / 2a ) x + ( b / 2a )² = ( x + b / 2a )²
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Thank you so much!!! :)
WellHello!
To rewrite the quadratic function f(x) = ax^2 + bx + c in completed square form, we need to complete the square by adding and subtracting a term. Here's how you can do it step by step:
Step 1: Start with the quadratic function f(x) = ax^2 + bx + c.
Step 2: Group the quadratic terms (ax^2 and bx) together and leave the constant term (c) separate.
f(x) = (ax^2 + bx) + c
Step 3: Factor out the coefficient of x^2 from the quadratic terms. (Take out common factor 'a' from ax^2 and bx)
f(x) = a(x^2 + (b/a)x) + c
Step 4: To complete the square, take half of the coefficient of 'x', square it, and add/subtract the result within the parentheses. The value to add/subtract is (b/2a)^2.
f(x) = a(x^2 + (b/a)x + (b/2a)^2 - (b/2a)^2) + c
Step 5: Simplify the quadratic expression inside the parentheses.
f(x) = a(x^2 + (b/a)x + (b/2a)^2 - (b/2a)^2) + c
Step 6: Simplify the expression (b/2a)^2 - (b/2a)^2.
Since it is a subtraction, the terms cancel out, resulting in zero.
f(x) = a(x^2 + (b/a)x + 0) + c
Step 7: Remove the parentheses.
f(x) = a(x^2 + (b/a)x) + c
Step 8: Finally, factor the quadratic expression inside the parentheses.
f(x) = a(x + (b/2a))^2 + c
So, the quadratic function f(x) = ax^2 + bx + c in completed square form is f(x) = a(x + (b/2a))^2 + c.
Note: The completed square form helps in identifying the vertex of a parabola, which is (-b/2a, c).