What are the roots of the expression
𝑥
2
−
7
𝑥
+
6
=
0
?

Question

What are the roots of the expression 
𝑥
2
−
7
𝑥
+
6
=
0
?

BlackTom AI · Accepted Answer

Two numbers that multiply to +6 and add to −7 are −1 and −6, so the quadratic can be factored.

1. Relevant knowledge points and concepts
- A quadratic ax^2 + bx + c = 0 can be solved by factoring if two numbers p and q exist with p·q = a·c and p + q = b (for a = 1, simply p·q = c and p + q = b).
- The Zero Product Property: if (x − r)(x − s) = 0, then x = r or x = s.
- The quadratic formula x = [−b ± sqrt(b^2 − 4ac)]/(2a) is a general method and provides a check.

2. Complete reasoning process (step by step)
Start with x^2 − 7x + 6 = 0.
- Find two numbers whose product is c = +6 and whose sum is b = −7. Those numbers are −1 and −6 because (−1)·(−6) = +6 and (−1) + (−6) = −7.
- Rewrite the quadratic as x^2 − x − 6x + 6 and factor by grouping:
  x(x − 1) − 6(x − 1) = (x − 1)(x − 6).
- Apply the Zero Product Property: (x − 1)(x − 6) = 0 implies x − 1 = 0 or x − 6 = 0.
- Solve: x = 1 or x = 6.

3. Calculations and intermediate steps (quadratic formula check)
Using a = 1, b = −7, c = 6:
- Discriminant D = b^2 − 4ac = (−7)^2 − 4(1)(6) = 49 − 24 = 25.
- x = [−b ± sqrt(D)]/(2a) = [7 ± 5]/2.
- Thus x = (7 + 5)/2 = 12/2 = 6, or x = (7 − 5)/2 = 2/2 = 1.

4. Why the correct answer is correct
Factoring directly yields (x − 1)(x − 6) = 0, and both factoring and the quadratic formula give x = 1 and x = 6. Substitution verifies: 1^2 − 7·1 + 6 = 0 and 6^2 − 7·6 + 6 = 0.

5. Why other potential options are incorrect
Common incorrect guesses like x = −1 or x = −6 fail because they do not satisfy the equation (e.g., (−1)^2 − 7(−1) + 6 = 1 + 7 + 6 ≠ 0). Any root must satisfy both the sum and product conditions (sum = 7, product = 6 for the positive root signs), which only 1 and 6 do here.

6. Conclusion
Therefore the roots are x = 1 and x = 6, which matches the provided answer.

What are the roots of the expression 𝑥 2 − 7 𝑥 + 6 = 0 ?Single choice

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