Question

Solve the quadratic equation X^2+5x+6using the completing the square method. Be sure to explain each step.

Answer 1

Explanation

We must solve the following quadratic equation by completing squares:

$x^2+5x+6=0.$

1) First, we rewrite the equation in the following way:

$x^2+2\cdot\frac{5}{2}\cdot x+6=0.$

2) We sum to both sides of the equation (5/2)², get:

$x^2+2\cdot\frac{5}{2}\cdot x+(\frac{5}{2})^2+6=(\frac{5}{2})^2.$

3) Expressing the first term as a square, we have:

$(x+\frac{5}{2})^2+6=\frac{25}{4}.$

4) Passing the +6 at the left as -6 at the right:

$\begin{gathered} (x+\frac{5}{2})^2=\frac{25}{4}-6, \\ (x+\frac{5}{2})^2=\frac{25}{4}-\frac{24}{4}, \\ (x+\frac{5}{2})^2=\frac{1}{4}. \end{gathered}$

5) By taking the square root on both sides, we get two solutions:

$\begin{gathered} x+\frac{5}{2}=+\frac{1}{2}\rightarrow x=\frac{1}{2}-\frac{5}{2}=-\frac{4}{2}=-2, \\ x+\frac{5}{2}=-\frac{1}{2}\rightarrow x=-\frac{1}{2}-\frac{5}{2}=-\frac{6}{2}=-3. \end{gathered}$Answer

1) Re-writing the equation:

$x^2+2\cdot\frac{5}{2}\cdot x+6=0.$

2) Summing (5/2)² on both sides:

$x^2+2\cdot\frac{5}{2}\cdot x+(\frac{5}{2})^2+6=(\frac{5}{2})^2.$

3) Expressing the first term as a square:

$(x+\frac{5}{2})^2+6=\frac{25}{4}.$

4) Simplifying:

$\begin{gathered} (x+\frac{5}{2})^2=\frac{25}{4}-6, \\ (x+\frac{5}{2})^2=\frac{25}{4}-\frac{24}{4}, \\ (x+\frac{5}{2})^2=\frac{1}{4}. \end{gathered}$

5) Taking the square root:

$\begin{gathered} x+\frac{5}{2}=+\frac{1}{2}\rightarrow x=\frac{1}{2}-\frac{5}{2}=-\frac{4}{2}=-2, \\ x+\frac{5}{2}=-\frac{1}{2}\rightarrow x=-\frac{1}{2}-\frac{5}{2}=-\frac{6}{2}=-3. \end{gathered}$