What are the roots of the expression
𝑥
2
+
5
𝑥
=
0
?

Question

What are the roots of the expression 
𝑥
2
+
5
𝑥
=
0
?

BlackTom AI · Accepted Answer

Factorization: x^2 + 5x = 0 can be solved by factoring because each term contains x.

1. Relevant knowledge points and concepts
- Factoring common factors from polynomials.
- Zero-Product Property: if a·b = 0, then a = 0 or b = 0.
- As a check, the quadratic formula x = [-b ± sqrt(b^2 - 4ac)]/(2a) applies to ax^2 + bx + c = 0.

2. Step-by-step reasoning
Start with the equation:
x^2 + 5x = 0

Factor the common factor x from both terms:
x(x + 5) = 0

Apply the Zero-Product Property: set each factor equal to zero:
x = 0  or  x + 5 = 0

Solve the second equation:
x + 5 = 0  ⇒  x = −5

Thus the two solutions are x = 0 and x = −5.

3. Intermediate calculations (shown explicitly)
- Factoring: x^2 + 5x = x(x + 5)
- Zero-product split: x = 0; x + 5 = 0 ⇒ x = −5

Alternative check with quadratic formula (a = 1, b = 5, c = 0):
x = [-5 ± sqrt(5^2 − 4·1·0)]/(2·1)
x = [-5 ± sqrt(25)]/2
x = [-5 ± 5]/2
Two cases:
(-5 + 5)/2 = 0/2 = 0
(-5 − 5)/2 = −10/2 = −5

4. Why the correct answer is correct
Both methods give the same pair of roots because factoring correctly exposes the factor x, so the equation is satisfied when x = 0 or when x = −5.

5. Why other answers would be incorrect
Common mistakes include dividing both sides by x (which would lose the x = 0 solution), misfactoring as x(x − 5) (which would give x = 5 instead of −5), or arithmetic sign errors in the quadratic formula. Any answer other than x = 0 and x = −5 fails to satisfy the original equation when substituted back.

6. Conclusion
The roots are x = 0 and x = −5, which matches the provided answer.

What are the roots of the expression 𝑥 2 + 5 𝑥 = 0 ?单项选择题

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