Sure, let's solve the given quadratic equation step by step. The equation we have is:
[tex]\[ y = x^2 - 5x - 14 \][/tex]
### Step 1: Factoring the quadratic expression
To factor the quadratic expression [tex]\( x^2 - 5x - 14 \)[/tex], we need to find two numbers that multiply to [tex]\(-14\)[/tex] (the constant term) and add up to [tex]\(-5\)[/tex] (the coefficient of [tex]\(x\)[/tex]).
List the pairs of factors of [tex]\(-14\)[/tex]:
- [tex]\((-1, 14)\)[/tex]
- [tex]\((1, -14)\)[/tex]
- [tex]\((2, -7)\)[/tex]
- [tex]\((-2, 7)\)[/tex]
Among these pairs, we need to find the one that adds up to [tex]\(-5\)[/tex]:
- [tex]\((-1) + 14 = 13\)[/tex]
- [tex]\( 1 + (-14) = -13\)[/tex]
- [tex]\( 2 + (-7) = -5\)[/tex]
- [tex]\((-2) + 7 = 5\)[/tex]
The pair [tex]\((2, -7)\)[/tex] adds up to [tex]\(-5\)[/tex], which is what we need.
### Step 2: Writing the equation in factored form
Now that we have found the pair of numbers [tex]\(2\)[/tex] and [tex]\(-7\)[/tex], we write the quadratic equation in its factored form:
[tex]\[ y = (x + 2)(x - 7) \][/tex]
### Step 3: Finding the zeros/solutions of the equation
To find the zeros or [tex]\(x\)[/tex]-intercepts of the equation, we set each factor equal to zero and solve for [tex]\(x\)[/tex]:
1. [tex]\( x + 2 = 0 \)[/tex]
[tex]\[ x = -2 \][/tex]
2. [tex]\( x - 7 = 0 \)[/tex]
[tex]\[ x = 7 \][/tex]
### Final Answer
So, the quadratic equation [tex]\( y = x^2 - 5x - 14 \)[/tex] can be factored as [tex]\( y = (x + 2)(x - 7) \)[/tex], and its zeros/solutions are:
[tex]\[ x = -2 \quad \text{and} \quad x = 7 \][/tex]
