### Home > INT2 > Chapter 12 > Lesson 12.1.2 > Problem12-24

12-24.

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

1. Complete the square to write the equation in graphing form.

Use the Math Notes box.

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.

 $2.5$$+$$x$ $2.5x$ $=-4$ $x^2$ $2.5x$ $x+2.5$

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, $x+2.5=\underline{+}1.5$. 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. What does the graph tell you about the solutions to the equation $x^2 + 6x + 11 = 0$? Explain.

Are there any $x$-intercepts?