MATH SOLVE

3 months ago

Q:
# Complete the square to find the minimum value of the expression 4x2 + 8x + 23.

Accepted Solution

A:

So you need to come up with a perfect square that works for the x coefficients.

like.. (2x + 2)^2

(2x+2)(2x+2) = 4x^2 + 8x + 4

Compare this to the equation given. Our perfect square has +4 instead of +23. The difference is: 23 - 4 = 19

I'm going to assume the given equation equals zero..

So, If we add subtract 19 from both sides of the equation we get the perfect square.

4x^2 + 8x + 23 - 19 = 0 - 19

4x^2 + 8x + 4 = - 19

complete the square and move 19 over..

(2x+2)^2 + 19 = 0

factor the 2 out becomes 2^2 = 4

ANSWER: 4(x+1)^2 + 19 = 0

for a short cut, the standard equation

ax^2 + bx + c = 0 becomes a(x - h)^2 + k = 0

Where "a, b, c" are the same and ..

h = -b/(2a)

k = c - b^2/(4a)

Vertex = (h, k)

this will be a minimum point when "a" is positive upward facing parabola and a maximum point when "a" is negative downward facing parabola.

like.. (2x + 2)^2

(2x+2)(2x+2) = 4x^2 + 8x + 4

Compare this to the equation given. Our perfect square has +4 instead of +23. The difference is: 23 - 4 = 19

I'm going to assume the given equation equals zero..

So, If we add subtract 19 from both sides of the equation we get the perfect square.

4x^2 + 8x + 23 - 19 = 0 - 19

4x^2 + 8x + 4 = - 19

complete the square and move 19 over..

(2x+2)^2 + 19 = 0

factor the 2 out becomes 2^2 = 4

ANSWER: 4(x+1)^2 + 19 = 0

for a short cut, the standard equation

ax^2 + bx + c = 0 becomes a(x - h)^2 + k = 0

Where "a, b, c" are the same and ..

h = -b/(2a)

k = c - b^2/(4a)

Vertex = (h, k)

this will be a minimum point when "a" is positive upward facing parabola and a maximum point when "a" is negative downward facing parabola.