Answer :
Certainly! Let's find the roots of the quadratic equation [tex]\( f(x) = x^2 - x - 20 \)[/tex] in three different ways: using the quadratic formula, by factoring, and through symbolic computation.
### Method 1: Using the Quadratic Formula
The quadratic formula for finding the roots of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For [tex]\( f(x) = x^2 - x - 20 \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -1 \)[/tex]
- [tex]\( c = -20 \)[/tex]
Let's apply the quadratic formula:
1. Calculate the discriminant [tex]\( \Delta \)[/tex]:
[tex]\[ \Delta = b^2 - 4ac = (-1)^2 - 4(1)(-20) = 1 + 80 = 81 \][/tex]
2. Find the roots:
[tex]\[ x = \frac{-(-1) \pm \sqrt{81}}{2(1)} = \frac{1 \pm 9}{2} \][/tex]
So the roots are:
[tex]\[ \begin{cases} x_1 = \frac{1 + 9}{2} = \frac{10}{2} = 5 \\ x_2 = \frac{1 - 9}{2} = \frac{-8}{2} = -4 \end{cases} \][/tex]
Thus, the roots are [tex]\( x_1 = 5 \)[/tex] and [tex]\( x_2 = -4 \)[/tex].
### Method 2: Factoring the Quadratic Equation
To factor the quadratic equation [tex]\( f(x) = x^2 - x - 20 \)[/tex], we need to find two numbers that multiply to [tex]\( -20 \)[/tex] and add to [tex]\( -1 \)[/tex].
The factors of [tex]\( -20 \)[/tex] that add up to [tex]\( -1 \)[/tex] are [tex]\( 5 \)[/tex] and [tex]\( -4 \)[/tex].
So, we can write:
[tex]\[ x^2 - x - 20 = (x - 5)(x + 4) \][/tex]
Setting each factor to zero gives the roots:
[tex]\[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \][/tex]
[tex]\[ x + 4 = 0 \quad \Rightarrow \quad x = -4 \][/tex]
Thus, the roots are [tex]\( x = 5 \)[/tex] and [tex]\( x = -4 \)[/tex].
### Method 3: Symbolic Computation
We can use symbolic computation to find the roots. For the quadratic equation [tex]\( f(x) = x^2 - x - 20 \)[/tex]:
1. Set up the equation [tex]\( x^2 - x - 20 = 0 \)[/tex].
2. Solve for [tex]\( x \)[/tex] symbolically.
The roots are:
[tex]\[ x = 5 \text{ and } x = -4 \][/tex]
### Summary of Results
The roots of the quadratic equation [tex]\( f(x) = x^2 - x - 20 \)[/tex] are:
- By the quadratic formula: [tex]\( x_1 = 5 \)[/tex] and [tex]\( x_2 = -4 \)[/tex]
- By factoring: [tex]\( x = 5 \)[/tex] and [tex]\( x = -4 \)[/tex]
- By symbolic computation: [tex]\( x = 5 \)[/tex] and [tex]\( x = -4 \)[/tex]
Each method confirms that the roots of [tex]\( f(x) = x^2 - x - 20 \)[/tex] are [tex]\( x_1 = 5 \)[/tex] and [tex]\( x_2 = -4 \)[/tex].
### Method 1: Using the Quadratic Formula
The quadratic formula for finding the roots of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For [tex]\( f(x) = x^2 - x - 20 \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -1 \)[/tex]
- [tex]\( c = -20 \)[/tex]
Let's apply the quadratic formula:
1. Calculate the discriminant [tex]\( \Delta \)[/tex]:
[tex]\[ \Delta = b^2 - 4ac = (-1)^2 - 4(1)(-20) = 1 + 80 = 81 \][/tex]
2. Find the roots:
[tex]\[ x = \frac{-(-1) \pm \sqrt{81}}{2(1)} = \frac{1 \pm 9}{2} \][/tex]
So the roots are:
[tex]\[ \begin{cases} x_1 = \frac{1 + 9}{2} = \frac{10}{2} = 5 \\ x_2 = \frac{1 - 9}{2} = \frac{-8}{2} = -4 \end{cases} \][/tex]
Thus, the roots are [tex]\( x_1 = 5 \)[/tex] and [tex]\( x_2 = -4 \)[/tex].
### Method 2: Factoring the Quadratic Equation
To factor the quadratic equation [tex]\( f(x) = x^2 - x - 20 \)[/tex], we need to find two numbers that multiply to [tex]\( -20 \)[/tex] and add to [tex]\( -1 \)[/tex].
The factors of [tex]\( -20 \)[/tex] that add up to [tex]\( -1 \)[/tex] are [tex]\( 5 \)[/tex] and [tex]\( -4 \)[/tex].
So, we can write:
[tex]\[ x^2 - x - 20 = (x - 5)(x + 4) \][/tex]
Setting each factor to zero gives the roots:
[tex]\[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \][/tex]
[tex]\[ x + 4 = 0 \quad \Rightarrow \quad x = -4 \][/tex]
Thus, the roots are [tex]\( x = 5 \)[/tex] and [tex]\( x = -4 \)[/tex].
### Method 3: Symbolic Computation
We can use symbolic computation to find the roots. For the quadratic equation [tex]\( f(x) = x^2 - x - 20 \)[/tex]:
1. Set up the equation [tex]\( x^2 - x - 20 = 0 \)[/tex].
2. Solve for [tex]\( x \)[/tex] symbolically.
The roots are:
[tex]\[ x = 5 \text{ and } x = -4 \][/tex]
### Summary of Results
The roots of the quadratic equation [tex]\( f(x) = x^2 - x - 20 \)[/tex] are:
- By the quadratic formula: [tex]\( x_1 = 5 \)[/tex] and [tex]\( x_2 = -4 \)[/tex]
- By factoring: [tex]\( x = 5 \)[/tex] and [tex]\( x = -4 \)[/tex]
- By symbolic computation: [tex]\( x = 5 \)[/tex] and [tex]\( x = -4 \)[/tex]
Each method confirms that the roots of [tex]\( f(x) = x^2 - x - 20 \)[/tex] are [tex]\( x_1 = 5 \)[/tex] and [tex]\( x_2 = -4 \)[/tex].
