Find the roots of x2 + 4x – 5 = 0 by factorisation.
The correct answer and explanation is:
To find the roots of the quadratic equation
x² + 4x − 5 = 0
by factorisation, we follow these steps:
✅ Correct Answer:
Factorising:
x² + 4x − 5 = (x + 5)(x − 1) = 0
Setting each factor equal to zero:
x + 5 = 0 → x = −5
x − 1 = 0 → x = 1
∴ The roots are x = −5 and x = 1.
🧠 300 Words Explanation:
Factorisation is a method used to break a quadratic expression into two binomial expressions whose product gives the original quadratic. The general form of a quadratic is:
ax² + bx + c = 0, where a, b, c are constants.
In our equation:
x² + 4x − 5 = 0,
we identify:
- a = 1
- b = 4
- c = −5
The goal is to split the middle term (4x) into two terms whose coefficients multiply to give a × c = 1 × (−5) = −5 and add up to b = 4.
We look for two numbers whose:
- Product = −5
- Sum = 4
These numbers are 5 and −1, because:
- 5 × (−1) = −5
- 5 + (−1) = 4
Now rewrite the equation using these two numbers:
x² + 5x − x − 5 = 0
Group the terms:
(x² + 5x) − (x + 5) = 0
Factor each group:
x(x + 5) −1(x + 5) = 0
Now factor the common binomial:
(x − 1)(x + 5) = 0
Use the zero product property (if ab = 0, then either a = 0 or b = 0):
- x − 1 = 0 → x = 1
- x + 5 = 0 → x = −5
Hence, the equation x² + 4x − 5 = 0 has two real roots:
x = −5 and x = 1
This shows that the equation can be solved purely by factoring and applying basic algebraic principles.