Question

select the correct answer.

if the factors of quadratic equation are \\((x - 7)\\) and \\((x + 3)\\), what are the \\(x\\)-intercepts?

\\((-3,0)\\) and \\((7,0)\\)
\\((3,0)\\) and \\((7,0)\\)
\\((-7,0)\\) and \\((-3,0)\\)
\\((-7,0)\\) and \\((3,0)\\)

Explanation:

🆕 New Concept Discovered: Finding x-intercepts from Factors
Finding where a graph crosses the x-axis

Step 1: Understand x-intercepts

An \(x\)-intercept is a point on the graph where the \(y\)-value is equal to \(0\). To find the \(x\)-intercepts of a quadratic equation, we set the factored equation equal to zero:

$$ (x - 7)(x + 3) = 0 $$

Step 2: Solve for x

Using the Zero Product Property, if the product of two factors is zero, at least one of the factors must be zero. We set each factor to \(0\) and solve for \(x\):

1. Set the first factor to zero:

$$ x - 7 = 0 \implies x = 7 $$

1. Set the second factor to zero:

$$ x + 3 = 0 \implies x = -3 $$

Step 3: Write as coordinate points

Since \(x\)-intercepts are points on a graph, they are written in the coordinate form \((x, y)\) where \(y = 0\):

• For \(x = 7\), the point is \((7, 0)\).
• For \(x = -3\), the point is \((-3, 0)\).

Answer:

(-3,0) and (7,0) (First option)