To find the [tex]\( x \)[/tex]-intercepts of the graph of the quadratic equation [tex]\( y = x^2 + 7x + 12 \)[/tex], we need to determine the values of [tex]\( x \)[/tex] for which [tex]\( y = 0 \)[/tex]. In other words, we need to solve the equation [tex]\( x^2 + 7x + 12 = 0 \)[/tex].
Here's a step-by-step solution:
1. Identify the quadratic equation:
[tex]\[
x^2 + 7x + 12 = 0
\][/tex]
2. Factorize the quadratic equation:
To factor the quadratic equation, we need to find two numbers that multiply to give the constant term (12) and add to give the coefficient of the linear term (7).
The pairs of numbers that multiply to 12 are:
[tex]\[
(1, 12),\ (2, 6),\ (3, 4),\ (-1, -12),\ (-2, -6),\ (-3, -4)
\][/tex]
Among these pairs, the pair that adds up to 7 is:
[tex]\[
(3, 4)
\][/tex]
3. Write the equation in its factored form:
[tex]\[
x^2 + 7x + 12 = (x + 3)(x + 4) = 0
\][/tex]
4. Set each factor equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
x + 3 = 0 \quad \Rightarrow \quad x = -3
\][/tex]
[tex]\[
x + 4 = 0 \quad \Rightarrow \quad x = -4
\][/tex]
5. Determine the [tex]\( x \)[/tex]-intercepts:
Each solution corresponds to an [tex]\( x \)[/tex]-intercept, as the [tex]\( y \)[/tex]-coordinate of an [tex]\( x \)[/tex]-intercept is always 0. Therefore, the [tex]\( x \)[/tex]-intercepts are:
[tex]\[
(-3, 0) \quad \text{and} \quad (-4, 0)
\][/tex]
Given the [tex]\( x \)[/tex]-intercepts, we can now match them to the provided choices:
- Choice A: [tex]\( (7,0) \)[/tex] and [tex]\( (12,0) \)[/tex]
- Choice B: [tex]\( (-7,0) \)[/tex] and [tex]\( (-12,0) \)[/tex]
- Choice C: [tex]\( (-4,0) \)[/tex] and [tex]\( (-3,0) \)[/tex]
- Choice D: [tex]\( (4,0) \)[/tex] and [tex]\( (3,0) \)[/tex]
The correct choice is:
C. [tex]\( (-4,0) \)[/tex] and [tex]\( (-3,0) \)[/tex].
Thus, the [tex]\( x \)[/tex]-intercepts of the graph of [tex]\( y = x^2 + 7x + 12 \)[/tex] are indeed [tex]\( (-4, 0) \)[/tex] and [tex]\( (-3, 0) \)[/tex].
