Question

Rewrite the function by completing the square. f(x)= 2x^{2}+3x-2f(x)=2x 2 +3x−2

Answer 1

Answer:

$2\Bigl[(x+\frac{3}{4} )^2-\frac{25}{16} \Bigr]$

Step-by-step explanation:

Step 1

We factor out 2 so that the coefficient of the quadratic term is 1.

$f(x)=2x^2+3x-2\\f(x)=2\bigl(x^2+\frac{3}{2} -1\bigr)$

Step 2

In this step we add and subtract  the square of the coefficient of the x term, this term is  $\Bigl(\frac{1}{2} \bigl(\frac{3}{2}\bigr)\Bigr)^2$. This is the step where we complete the square.

$f(x)=2\Bigl[x^2+\frac{3}{2}x+(\frac{3}{4})^2-(\tfrac{3}{4})^2 -1 \Bigr]\\f(x)=2\Bigl[x^2+\frac{3}{2}x+(\frac{3}{4})^2-\frac{25}{16} \Bigr]$

Step 3

In this step we factor out the perfect square tri-nomial formed by the first 3 terms in last line of step 2.  This calculation is shown below,

$f(x)=2\Bigl[\bigl(x+\frac{3}{4} \bigr)^2-\frac{25}{16} \Bigr]$