### Home > CCA > Chapter 9 > Lesson 9.3.1 > Problem9-76

9-76.

For the quadratic function $f(x)=x^2+6x+11$:

1. Use the idea of completing the square to write it in graphing form.

Example 3: Solve $x^2+5x+4=0$ by completing the square.

Solution: This method works most efficiently when the coefficient of $x^2$ is $1$.  Rewrite the equation as $x^2+5x=−4$.  Rewrite the left side as an incomplete square:

 Generic rectangle left edge$2.5$+$x$ interior top left$2.5x$ interior bottom left$x^2$ interior bottom right$2.5x$ $=-4$ bottom edge$x$    +    $2.5$

Complete the square and rewrite as
$(x+2.5)^2−6.25=−4$ or $(x+2.5)^2=2.25$

Take the square root of both sides, $x+2.5=\pm1.5$.  Solving for $x$ reveals that $x=−1$ or $x=−4$.

2. State the vertex and sketch a graph of the parabola.

In the standard equation, the vertex is ($h,k$).

$(−3,2)$

3. Use the graph to explain why the equation $x^2+6x+11=0$ has no real solutions.

Are there any $x$-intercepts?