Question

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

Answer 1

Step-by-step 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(5)/(2)\cdot x+6=0.`

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

`x^2+2\cdot(5)/(2)\cdot x+((5)/(2))^2+6=((5)/(2))^2.`

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

`(x+(5)/(2))^2+6=(25)/(4).`

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

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

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

`\begin{gathered} x+(5)/(2)=+(1)/(2)\rightarrow x=(1)/(2)-(5)/(2)=-(4)/(2)=-2, \\ x+(5)/(2)=-(1)/(2)\rightarrow x=-(1)/(2)-(5)/(2)=-(6)/(2)=-3. \end{gathered}`Answer

1) Re-writing the equation:

`x^2+2\cdot(5)/(2)\cdot x+6=0.`

2) Summing (5/2)² on both sides:

`x^2+2\cdot(5)/(2)\cdot x+((5)/(2))^2+6=((5)/(2))^2.`

3) Expressing the first term as a square:

`(x+(5)/(2))^2+6=(25)/(4).`

4) Simplifying:

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

5) Taking the square root:

`\begin{gathered} x+(5)/(2)=+(1)/(2)\rightarrow x=(1)/(2)-(5)/(2)=-(4)/(2)=-2, \\ x+(5)/(2)=-(1)/(2)\rightarrow x=-(1)/(2)-(5)/(2)=-(6)/(2)=-3. \end{gathered}`