Question
Factor the following quadratic equation: f(x)=x2−5x−6 . Then, verify the answer in a graph if possible.(1 point) Responses (x+1)(x+6) left parenthesis x plus 1 right parenthesis left parenthesis x plus 6 right parenthesis (x−1)(x+6) left parenthesis x minus 1 right parenthesis left parenthesis x plus 6 right parenthesis (x−1)(x−6) left parenthesis x minus 1 right parenthesis left parenthesis x minus 6 right parenthesis (x+1)(x−6)
Answers
Answered by
GPT-4o mini
To factor the quadratic equation \( f(x) = x^2 - 5x - 6 \), we need to find two numbers that multiply to \(-6\) (the constant term) and add up to \(-5\) (the coefficient of the linear term).
The numbers that satisfy these conditions are \(-6\) and \(1\). This is because:
- \(-6 \times 1 = -6\)
- \(-6 + 1 = -5\)
Thus, we can factor the quadratic as follows:
\[
f(x) = (x - 6)(x + 1)
\]
Now let's verify this factorization by expanding it back:
\[
(x - 6)(x + 1) = x^2 + x - 6x - 6 = x^2 - 5x - 6
\]
This matches the original function, confirming the factorization is correct.
### Graphical Verification
To further verify this, you can graph the function \( f(x) = x^2 - 5x - 6 \) and observe where the graph intersects the x-axis. The x-intercepts should correspond to the roots of the equation, which are \( x = 6 \) and \( x = -1 \).
Using a graphing tool or software:
1. Graph \( f(x) = x^2 - 5x - 6 \).
2. Look for the points where the graph intersects the x-axis. These points should be \( (6, 0) \) and \( (-1, 0) \).
Thus, the correct factored form is \( (x - 6)(x + 1) \) and the graphical representation will confirm the roots.
The numbers that satisfy these conditions are \(-6\) and \(1\). This is because:
- \(-6 \times 1 = -6\)
- \(-6 + 1 = -5\)
Thus, we can factor the quadratic as follows:
\[
f(x) = (x - 6)(x + 1)
\]
Now let's verify this factorization by expanding it back:
\[
(x - 6)(x + 1) = x^2 + x - 6x - 6 = x^2 - 5x - 6
\]
This matches the original function, confirming the factorization is correct.
### Graphical Verification
To further verify this, you can graph the function \( f(x) = x^2 - 5x - 6 \) and observe where the graph intersects the x-axis. The x-intercepts should correspond to the roots of the equation, which are \( x = 6 \) and \( x = -1 \).
Using a graphing tool or software:
1. Graph \( f(x) = x^2 - 5x - 6 \).
2. Look for the points where the graph intersects the x-axis. These points should be \( (6, 0) \) and \( (-1, 0) \).
Thus, the correct factored form is \( (x - 6)(x + 1) \) and the graphical representation will confirm the roots.