Answer :
Sure, let's go through the detailed step-by-step solution for the quadratic equation [tex]\(f(x) = x^2 - x - 6\)[/tex].
### Step 1: Understanding the given quadratic equation
We start with the quadratic equation:
[tex]\[ f(x) = x^2 - x - 6 \][/tex]
### Step 2: Factoring the quadratic equation
To find the factors of [tex]\(x^2 - x - 6\)[/tex], we look for two binomials that multiply together to give the original quadratic expression. The quadratic expression can be factored into:
[tex]\[ f(x) = (x - 3)(x + 2) \][/tex]
### Step 3: Verification of the factors
To ensure correctness, we can expand the binomials:
[tex]\[ (x - 3)(x + 2) = x(x + 2) - 3(x + 2) \][/tex]
[tex]\[ = x^2 + 2x - 3x - 6 \][/tex]
[tex]\[ = x^2 - x - 6 \][/tex]
This confirms that our factored form is correct:
[tex]\[ f(x) = (x - 3)(x + 2) \][/tex]
### Step 4: Finding the zeros of the function
The zeros of [tex]\(f(x)\)[/tex] are the values of [tex]\(x\)[/tex] that make [tex]\(f(x) = 0\)[/tex].
Set the factored form equal to zero:
[tex]\[ (x - 3)(x + 2) = 0 \][/tex]
Using the zero-product property, we set each factor equal to zero:
[tex]\[ x - 3 = 0 \][/tex]
[tex]\[ x = 3 \][/tex]
[tex]\[x + 2 = 0 \][/tex]
[tex]\[ x = -2 \][/tex]
### Step 5: Graphical interpretation
The solutions to the equation [tex]\( f(x) = 0 \)[/tex] are [tex]\( x = 3 \)[/tex] and [tex]\( x = -2 \)[/tex]. These are the points where the graph of the quadratic function crosses the x-axis.
### Summary
The quadratic function [tex]\( f(x) = x^2 - x - 6 \)[/tex] can be factored as [tex]\( (x - 3)(x + 2) \)[/tex]. The zeros of the function, where the graph intersects the x-axis, occur at [tex]\( x = 3 \)[/tex] and [tex]\( x = -2 \)[/tex].
Thus, we have the function in factorized form:
[tex]\[ f(x) = (x - 3)(x + 2) \][/tex]
And the zeros of the function are:
[tex]\[ x = 3 \][/tex]
[tex]\[ x = -2 \][/tex]
By following these steps, we successfully identified the factors and solutions of the quadratic equation.
### Step 1: Understanding the given quadratic equation
We start with the quadratic equation:
[tex]\[ f(x) = x^2 - x - 6 \][/tex]
### Step 2: Factoring the quadratic equation
To find the factors of [tex]\(x^2 - x - 6\)[/tex], we look for two binomials that multiply together to give the original quadratic expression. The quadratic expression can be factored into:
[tex]\[ f(x) = (x - 3)(x + 2) \][/tex]
### Step 3: Verification of the factors
To ensure correctness, we can expand the binomials:
[tex]\[ (x - 3)(x + 2) = x(x + 2) - 3(x + 2) \][/tex]
[tex]\[ = x^2 + 2x - 3x - 6 \][/tex]
[tex]\[ = x^2 - x - 6 \][/tex]
This confirms that our factored form is correct:
[tex]\[ f(x) = (x - 3)(x + 2) \][/tex]
### Step 4: Finding the zeros of the function
The zeros of [tex]\(f(x)\)[/tex] are the values of [tex]\(x\)[/tex] that make [tex]\(f(x) = 0\)[/tex].
Set the factored form equal to zero:
[tex]\[ (x - 3)(x + 2) = 0 \][/tex]
Using the zero-product property, we set each factor equal to zero:
[tex]\[ x - 3 = 0 \][/tex]
[tex]\[ x = 3 \][/tex]
[tex]\[x + 2 = 0 \][/tex]
[tex]\[ x = -2 \][/tex]
### Step 5: Graphical interpretation
The solutions to the equation [tex]\( f(x) = 0 \)[/tex] are [tex]\( x = 3 \)[/tex] and [tex]\( x = -2 \)[/tex]. These are the points where the graph of the quadratic function crosses the x-axis.
### Summary
The quadratic function [tex]\( f(x) = x^2 - x - 6 \)[/tex] can be factored as [tex]\( (x - 3)(x + 2) \)[/tex]. The zeros of the function, where the graph intersects the x-axis, occur at [tex]\( x = 3 \)[/tex] and [tex]\( x = -2 \)[/tex].
Thus, we have the function in factorized form:
[tex]\[ f(x) = (x - 3)(x + 2) \][/tex]
And the zeros of the function are:
[tex]\[ x = 3 \][/tex]
[tex]\[ x = -2 \][/tex]
By following these steps, we successfully identified the factors and solutions of the quadratic equation.
