How to Solve Quadratic Equations when a =
Rule of signs for real roots: If a and c have opposite signs, the 2 roots have opposite signs., Rule for the Diagonal Sum: Given a root pair of a quadratic equation (c1/a1), (c2/a2)., Rule., Examples of solving quadratic equations in the form x^2 +...
Step-by-Step Guide
-
Step 1: Rule of signs for real roots: If a and c have opposite signs
Example: the equation 6X^2
-11X
- 35 = 0 has 2 roots in opposite signs.
If a and c have same sign, the 2 roots have same sign If a and b have opposite signs, both roots are positive.
The equation: 21x^2
- 23x + 6 = 0 has 2 real roots, both positive.
If a and b have same sign, both roots are negative.
Example: the equation 15x^2 + 22x + 8 = 0 has 2 real roots, both negative. , Their product should equal to (c/a), meaning c = c1c2 and a = a1a2.
Their sum is: (c1/a1) + (c2/a2) = (c1a2 + c2a1)/a1a2 =
-b/a.
The sum c1a2 + c2a1 is called the Diagonal Sum of a root pair. , The Diagonal Sum of a TRUE root-pair must equal to (-b).
If it equals to (b), the answer is opposite in sign.
If a is negative, the above rule is reversal in sign. , In this case, the diagonal sum becomes the sum of the 2 real roots.
Solving results in finding 2 number knowing their sum (-b) and their product (c). -
Step 2: the 2 roots have opposite signs.
Solve: x^2
- 9x + 14 =
0.
Solution.
Both roots are positive.
Write down all factor-pairs of c =
14.
They are: (1, 14),(2, 7).
This second sum is: 2 + 7 = 9 =
-b.
The 2 real roots are: 2 and 7 , Solve: x^2 + 27x + 50 =
0.
Solution.
Rule of Signs shows that both roots are negative.
Write down all factor pairs of c = 50: (-1,
-50),(-2,
-25)...Stop! This sum is
-27 =
-b.
The 2 real roots are:
-2 and
-25. , Solve :
-x^2
- 26x + 56 =
0.
Solution:
Roots have opposite signs, a is negative.
Factor pairs of ac =
-56.
They are: (-1, 56),(-2, 28)...
Stop!.
This sum is: 28
- 2 = 26 =
-b.
According to the Rule when a is negative, the answer is opposite to the second set.
The real roots are: 2,
-28. , Solve: x^2 + 34x
- 72 =
0.
Solution.
Roots have opposite signs. a is positive.
Write down all factor pairs of
-72.
Stop when you find the sum = b (or
-b). (-1, 72), (-2, 36)...Stop!.
The sum of this set is 34 = b.
The answer is opposite to this set.
Two real roots are 2 and
-36. , When a = 1, solving quadratic equation in the form x^2 + bx + c = 0 by the Diagonal Sum Method is simple, fast and doesn't require factoring. -
Step 3: Rule for the Diagonal Sum: Given a root pair of a quadratic equation (c1/a1)
-
Step 4: (c2/a2).
-
Step 5: Examples of solving quadratic equations in the form x^2 + bx + c = 0.
-
Step 6: Example 1.
-
Step 7: Example 2.
-
Step 8: Example 3.
-
Step 9: Example 4.
-
Step 10: Conclusion.
Detailed Guide
Example: the equation 6X^2
-11X
- 35 = 0 has 2 roots in opposite signs.
If a and c have same sign, the 2 roots have same sign If a and b have opposite signs, both roots are positive.
The equation: 21x^2
- 23x + 6 = 0 has 2 real roots, both positive.
If a and b have same sign, both roots are negative.
Example: the equation 15x^2 + 22x + 8 = 0 has 2 real roots, both negative. , Their product should equal to (c/a), meaning c = c1c2 and a = a1a2.
Their sum is: (c1/a1) + (c2/a2) = (c1a2 + c2a1)/a1a2 =
-b/a.
The sum c1a2 + c2a1 is called the Diagonal Sum of a root pair. , The Diagonal Sum of a TRUE root-pair must equal to (-b).
If it equals to (b), the answer is opposite in sign.
If a is negative, the above rule is reversal in sign. , In this case, the diagonal sum becomes the sum of the 2 real roots.
Solving results in finding 2 number knowing their sum (-b) and their product (c).
Solve: x^2
- 9x + 14 =
0.
Solution.
Both roots are positive.
Write down all factor-pairs of c =
14.
They are: (1, 14),(2, 7).
This second sum is: 2 + 7 = 9 =
-b.
The 2 real roots are: 2 and 7 , Solve: x^2 + 27x + 50 =
0.
Solution.
Rule of Signs shows that both roots are negative.
Write down all factor pairs of c = 50: (-1,
-50),(-2,
-25)...Stop! This sum is
-27 =
-b.
The 2 real roots are:
-2 and
-25. , Solve :
-x^2
- 26x + 56 =
0.
Solution:
Roots have opposite signs, a is negative.
Factor pairs of ac =
-56.
They are: (-1, 56),(-2, 28)...
Stop!.
This sum is: 28
- 2 = 26 =
-b.
According to the Rule when a is negative, the answer is opposite to the second set.
The real roots are: 2,
-28. , Solve: x^2 + 34x
- 72 =
0.
Solution.
Roots have opposite signs. a is positive.
Write down all factor pairs of
-72.
Stop when you find the sum = b (or
-b). (-1, 72), (-2, 36)...Stop!.
The sum of this set is 34 = b.
The answer is opposite to this set.
Two real roots are 2 and
-36. , When a = 1, solving quadratic equation in the form x^2 + bx + c = 0 by the Diagonal Sum Method is simple, fast and doesn't require factoring.
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