Question

Complete the square to find the minimum value of the expression 4x^2 + 8x + 23 Minimum=???

Answer 1

We are given the following quadratic equation

$4x^2+8x+23$

Let us apply the completing the square method to the given equation.

Take the square of the half of the middle term coefficient (that is 8)

$(\frac{8}{2})^2=(4)^2=16$

Add and subtract this value from the given equation

$4x^2+8x+23+16-16$

Re-write the equation as below

$(4x^2+8x+16)+23-16$

The terms in the parenthesis will be a perfect square so we can factor it out as

$\begin{gathered} (4x^2+8x+16)+23-16 \\ \mleft(x+4\mright)^2+23-16 \\ \mleft(x+4\mright)^2+7 \end{gathered}$

Finally, compare this equation with the vertex form given by

$(x-h)^2+k$

So, we have

h = -4

k = 7

The vertex (h, k) = (-4, 7) is the minimum point of the given quadratic equation

$vertex=(-4,7)$