Complete the square to re-write the quadratic function in vertex form: y = 2x^2 + 8x + 4
Understand the Problem
The question is asking to complete the square for the given quadratic function in order to express it in vertex form. The approach involves manipulating the equation to isolate the perfect square trinomial.
Answer
The vertex form of the quadratic function is $$ y = 2(x + 2)^2 - 4 $$
Answer for screen readers
The vertex form of the quadratic function is
$$ y = 2(x + 2)^2 - 4 $$
Steps to Solve
- Factor out the coefficient of $x^2$
First, factor out the coefficient of $x^2$ (which is 2) from the terms involving $x$:
$$ y = 2(x^2 + 4x) + 4 $$
- Complete the square inside the parentheses
To complete the square, take the coefficient of $x$ (which is 4), divide it by 2 to get 2, and then square it to get 4. Add and subtract this value inside the parentheses:
$$ y = 2(x^2 + 4x + 4 - 4) + 4 $$
This rearranges to:
$$ y = 2((x + 2)^2 - 4) + 4 $$
- Simplify the equation
Distributing the 2 and simplifying the equation results in:
$$ y = 2(x + 2)^2 - 8 + 4 $$
Which simplifies further to:
$$ y = 2(x + 2)^2 - 4 $$
The vertex form of the quadratic function is
$$ y = 2(x + 2)^2 - 4 $$
More Information
Expressing a quadratic function in vertex form helps in identifying the vertex of the parabola represented by the function, which is useful for graphing and analyzing its properties.
Tips
- Forgetting to factor out the leading coefficient: Always remember to factor out any coefficient of $x^2$ before completing the square.
- Neglecting the signs while completing the square: When adding and subtracting the square, keep an eye on positive and negative signs.
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