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# Completing the Square

### Definition

For a quadratic polynomial $$f(x) = ax^2 + bx +c$$, completing the square means giving an expression of the form

$f(x) = A(x-B)^2 + C$

For example,

$-x^2 + 4x + 10 = - (x-2)^2 + 14$

We end up with a square term and a constant, both of which are easier for us to understand. This gives us an easy way to understand the graphs of quadratic polynomials, and to graph parabolas.

### 1. What is the maximum value of $$-x^2 + 4x + 10$$?

From above,

• $$-x^2 + 4x + 10 = - (x-2)^2 + 14$$.

Since squares are non-negative,

• $$- (x-2)^2 + 14 \leq 14$$.

Thus, the maximum value of the quadratic is $$14$$, which is achieved at $$x=2$$.

### 2. For what integer value $$n$$ is $$n^2 + 6x + 10$$ also a perfect square?

Completing the square, we see that

• $$n^2 + 6x + 10 = (n+3)^2 +1$$.

If

• $$n^2 + 6x + 10 = m^2$$

for some integer $$m$$, then

• $$1 = m^2 - (n+3)^2$$.

The only perfect squares that differ by $$1$$ are $$0$$ and $$1$$. Hence,

• $$(n+3)^2 = 0$$,

which has the solution $$n = -3$$.

Note by Arron Kau
3 years ago

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Could you explain how to make graph of y=f( k - x ) where y= $$ax^2 + bx +c$$ Or in that using the curve y=$$x^2 - 9x + 20$$. ? Where k is some constant. · 8 months ago