We will compare and contrast the four strategies for solving quadratic equations: completing the square, graphing, factoring, and using the discriminant. It will be easier to visualize these methods if we choose a quadratic equation to model them with, so let's use the following.
x^2-5x-7=0
Completing the Square
Completing the square is a technique for converting a quadratic polynomial of the form x^2+bx to the form (x-h)^2+k for some values of h and k. It can be done by adding ( b2)^2 to x^2+bx. Consider the given equation.
x^2-5x-7=0
First, let's add 7 to both sides to get all the terms with x on the left-hand side and the other terms on the right-hand side of the equation.
x^2-5x-7=0 ⇔ x^2-5x=7
For this equation, we have that b=- 5. Let's now calculate ( b2 )^2.
Next, we will add ( b2 )^2= 254 to both sides of our equation. Then we will factor the trinomial on the left-hand side using the formula for perfect square trinomials and solve the equation.
Both x= 52+ sqrt(53)2 and x= 52- sqrt(53)2 are solutions of the equation. Note that this method always give us the exact solutions.
Graphing
Now let's try graphing. In this method the first step is to plot the quadratic function, y=x^2-5x-7, on a coordinate plane.
The roots of a quadratic equation are the zeros of a quadratic function. Therefore, we should approximate where the graph crosses the x-axis.
Therefore, the roots of the equation x^2-5x-7=0 are approximately - 1.1 and 6.1. Note that by this method we are not able to find the exact solutions and usually we will spend a lot of time plotting the parabola in a coordinate system.
Factoring
To solve the given equation by factoring, we will start by identifying the values of a, b, and c.
x^2-5-7x=0 ⇕ 1x^2+( - 5)x+( - 7)=0
We have a quadratic equation with a= 1, b= - 5, and c= - 7. To factor the left-hand side we need to find a factor pair of 1 * ( - 7)=- 7 whose sum is - 5. Since - 7 is a negative number, we will only consider factors with opposite signs — one positive and one negative — so that their product is negative.
Factor Pair
Product of Factors
Sum of Factors
1 and - 7
^(1* (- 7)) - 7
1+(- 7) - 6
- 1 and 7
^(- 1* 7) - 7
- 1+7 6
Unfortunately, we did not find the integers whose product is - 7 and whose sum is - 5. Therefore, we cannot continue with this strategy. Factoring does not always work — especially when the roots are not rational numbers, like in our case. Note that if the coefficient c was a big number we must do a lot of calculations.
Discriminant
We will use the Quadratic Formula to solve the given quadratic equation.
ax^2+ bx+ c=0 ⇕ x=- b± sqrt(b^2-4 a c)/2 a
We first need to identify the values of a, b, and c.
x^2-5x-7=0 ⇕ 1x^2+( - 5)x+( - 7)=0
We see that a= 1, b= - 5, and c= - 7. Let's substitute these values into the Quadratic Formula.
The solutions for this equation are x= 5± sqrt(53)2. Let's separate them into the positive and negative cases.
x=5± sqrt(53)/2
x_1=5+sqrt(53)/2
x_2=5-sqrt(53)/2
Using the Quadratic Formula, we found that the solutions of the given equation are x_1= 5+sqrt(53)2 and x_2= 5-sqrt(53)2. Note that this method always gives us the exact solutions.
Conclusion
Now we will describe the advantages and disadvantages of each method. Unfortunately, not all the methods are equivalent.
Method
Type
Exact
Fast
Completing the Square
Algebraic
Yes
No
Graphing
Geometric
No
No
Factoring
Algebraic
No
No
Distriminant
Algebraic
Yes
Yes
In general, using discriminant is the best. It is usually the fastest and always exact.
