We will show how to solve the quadratic equation shown below by using three different methods: graphing, factoring, and using the Quadratic Formula. Then, we will decide which method works better for this case and why.
x^2+4x+4 = 0
a) Solving the Equation by Graphing
We can solve a quadratic equation of the form ax^2+bx+c = 0 by graphing its associated function f(x) = ax^2+bx+c. The x-intercepts of the graph are the solutions to the quadratic equation. Let's find the associated function for our equation.
&Original Equation &&Associated Function
& x^2+4x+4 = 0 && f(x) = x^2+4x+4
Now, we can graph the associated function to identify the solutions.
As we can see, the graph intersects the x-axis just once at (- 2, 0). Therefore, the equation has only one real solution, x=-2.
We can rewrite the new equation as shown below.
(x+2)^2 = 0 ⇔ (x- ( - 2))^2 = 0
By the Zero Product Property we can state that there is only one real solution, which is x= - 2.
c) Solving the Equation by Using the Quadratic Formula
Let's review the Quadratic Formula.
x = - b ± sqrt(b^2-4ac)/2a
To solve the quadratic equation by using the Quadratic Formula, we first need to identify the value of the parameters. We can do this by comparing our equation to the standard form of a quadratic equation.
ax^2+ bx+ c = 0
1x^2+ 4x+ 4 = 0
We can see that a= 1, b = 4 and c= 4. Now, we can substitute these values in the Quadratic Formula to solve the equation.
Once more, we have found that the equation has one real solution, x=- 2.
Conclusion
Preferences may vary, but for this case, solving by factoring is more convenient. The expression is easily factorable and this method does not require graphing, which can be complicated if we do not have a graphing calculator.
