Factorise x2 – 5x – 24, and hence solve x2 – 5x – 24 = 0.
The correct answer and explanation is:
Let’s start with the quadratic expression:
x2−5x−24x^2 – 5x – 24
Step 1: Factorise the quadratic
To factorise x2−5x−24x^2 – 5x – 24, look for two numbers that multiply to give the constant term, −24-24, and add to give the coefficient of the middle term, −5-5.
The pairs of numbers that multiply to −24-24 are:
- 11 and −24-24 (sum = −23-23)
- −1-1 and 2424 (sum = 2323)
- 22 and −12-12 (sum = −10-10)
- −2-2 and 1212 (sum = 1010)
- 33 and −8-8 (sum = −5-5)
- −3-3 and 88 (sum = 55)
The pair that adds up to −5-5 is 33 and −8-8, but since 3+(−8)=−53 + (-8) = -5, we take these numbers.
Thus, the quadratic can be factorised as: x2−5x−24=(x+3)(x−8)x^2 – 5x – 24 = (x + 3)(x – 8)
Step 2: Solve the equation x2−5x−24=0x^2 – 5x – 24 = 0
Since the quadratic factors as (x+3)(x−8)=0(x + 3)(x – 8) = 0, for the product to be zero, at least one of the factors must be zero. This gives two solutions:
- x+3=0x + 3 = 0
x=−3x = -3 - x−8=0x – 8 = 0
x=8x = 8
Summary
The factorised form of x2−5x−24x^2 – 5x – 24 is (x+3)(x−8)(x + 3)(x – 8). Solving x2−5x−24=0x^2 – 5x – 24 = 0 gives the solutions x=−3x = -3 and x=8x = 8.
Explanation
A quadratic equation like x2−5x−24x^2 – 5x – 24 represents a parabola when graphed. The factorisation method involves finding two numbers that multiply to the constant term and add to the coefficient of the middle term. This process breaks the quadratic into two binomials.
Once factorised, setting each binomial equal to zero uses the zero product property, which states that if the product of two expressions is zero, at least one of the expressions must be zero. This property helps find the roots or solutions of the equation.
The two solutions, x=−3x = -3 and x=8x = 8, are the x-values where the parabola crosses the x-axis. These points are also called the roots or zeros of the quadratic function.
This method is efficient for quadratics that factorise easily. If factorisation is difficult, other methods like completing the square or using the quadratic formula can be used.