If you are given a quadratic equation how do you determine which method to use to solve it?

Answer 1

You can always use the quadratic formula. However, some of the other methods are sometimes faster, if you know how to use them. Reminder: The quadratic equation has the form ax2 + bx + c = 0. I suggest you proceed in the following order:

• Convert the equation so that all coefficients are integers. For example, if there is a factor 1/2, multiply all terms by 2.
• If a is not equal to 1, best to use the quadratic formula.
• If b is even, complete the square.
• Otherwise (a = 1, b is odd), try factoring. If this doesn't work, use the quadratic equation.

(Improved Answer by Nghi1350)

The best method to select depends on the form of the given quadratic equation.

1. If a =1 - Solving the equation type x^2 + bx + c = 0.

The best and fastest method is the new Diagonal Sum method (Amazon e-book 2010). This method uses the Rule of Signs to know the signs of the 2 real roots before proceeding. It finds 2 numbers knowing their sum (-b) and their product (c). First, write down the factor-pairs of c. Stop when the sum of a pair is equal to (-b) or (b). There is no needs for factoring.

Example. Solve: x^2 - 9x - 102 = 0. The rule of signs indicates that the 2 roots have opposite signs (one - and one +). Write down factor-pairs of c = -102:

(-1, 102),(-2, 51),(-3, 34), (-6, 17)...Stop. This sum is 17 - 6 = 9 = -b. The 2 real roots are -6 and 17.

NOTE. There are opposite pairs (1, -102),(2, -51)...but they can be ignored since they are just the opposite of probable root pairs.

Example. Solve x^2 - 39x + 108 = 0. Both real roots are positive. Write factor pairs of c = 108: (1, 108)(2, 54)(3, 36)...Stop. This sum is 3 + 36 = 39 = -b. The 2 real roots are 3 and 36.

NOTE. If any pair's sum matches (-b) or (b), then we can conclude that this given quad. equation can't be factored, and consequently the quadratic formula must be used to solve it.

2. When a is not 1 - Solving equation ax^2 + bx + c = 0. The method to select depends on the values of a and c. The concept of this new method is direct finding the 2 real roots, in the form of 2 fractions, knowing their sum (-b/a) and product (c/a).

a. If a and c are both prime. In this case use the new Diagonal Sum method. This method directly select the probable root-pairs (in the form of 2 fractions) from the product of the 2 real roots (c/a). If a and c are prime there is always one unique root pair (except when 1 or -1 is a real root). If the diagonal sum of this probable root pair is not equal to (-b) or (b), then the equation can't be factored and you must use the quadratic formula.

Example. Solve 7x^2 - 76x - 11 = 0. Roots have opposite signs. There is unique probable root pair: (-1/7, 11/1). Its Diagonal Sum is: 77 - 1 = 76 = -b. The 2 real roots are -1/7 and 11/1.

NOTE. There is another solving method called the " factoring ac method" (You Tube) that you must know and study.

b. a and c are small numbers and may contain themselves one or 2 factors. In this case, use the new Diagonal Sum Method. The number of probable root-pairs is usually fewer than 3. You may calculate their diagonal sums and find the one that fits. Stop calculation when one diagonal sum matches b or -b.

Example. Solve: 8x^2 - 22x - 13 = 0. Roots have opposite signs. The (c/a) set up: (-1, 13)/[(1, 8)(2, 4)]. There are 3 probable root pairs: (-1/8, 13/1),(-1/2, 13/4),(-1/4, 13/2). The diagonal sum of the second pair is -4 + 26 = 22 = -b. The 2 real roots are -1/2 and 13/4. If any diagonal sum matches (b) or (-b), the equation can't be factored and you must use the quadratic formula.

c. a and c are large numbers and may contain themselves a few factors.

In this case, before proceeding with the Diagonal Sum method, you have to eliminate the factor-pairs that are not fitted. There are a few simple rules used to eliminate the no-fitted factor-pairs.

Example. Solve 45x^2 - 74x - 55 = 0. Roots have opposite signs. In order to do not omit any probable root pair, write down the (c/a) setup:

Numerator: Factor-pairs of c = -55: (-1, 55)(-5, 11)

Denominator: Factor pairs of a = 45: (1, 45)(3, 15)(5, 9)

First, you can eliminate the pair (-1, 55) of the numerator and the 2 pairs ((1, 45)(3, 15) of the denominator, since they give large diagonal sums as compared to (b = -74). The remainder (c/a): (-5, 11)/(5, 9) gives unique root pair:

(-5/9, 11/5). Its diagonal sum is -25 + 99 = 74 = -b. The 2 real roots are -5/9 and 11/5.

NOTE. When a and c are large numbers, the factoring ac method becomes inconvenient since the product ac is too large to handle.

Example. Solve: 12x^2 + 5x - 72 = 0. Roots have opposite signs. The c/a setup:

Numerator: (-1, 72)(-2, 36)(-3, 24)(-4, 18)(-6, 12)(-8, 9)

Denominator: ((1, 12)(2, 6)(3, 4)

First eliminate the pairs (-2, 36)(-4, 18)(-6, 12)/(2, 6) because they give even-number diagonal sums (while b = 5 is odd). Next, eliminate the pairs (-1, 72)(-3, 24)/(1, 12) because they give large number diagonal sums (while b = 5). The remainder c/a gives 2 probable root pairs: (-8/3, 9/4) and (-8/4, 9/3). The first diagonal sum is 27 - 32 = -5 = -b. The 2 real roots are -8/3 and 9/4.