What are the $x$-intercepts of the graph of $x^2+7x+10=0$?
The problem asks for the x-intercepts of the graph defined by the quadratic equation \(x^2 + 7x + 10 = 0\). An x-intercept occurs at a point \((x, 0)\) on the graph, which means the x-intercepts correspond to the roots (solutions) of the equation \(x^2 + 7x + 10 = 0\). Our goal is to solve for \(x\) and express the final result as a pair of coordinates.
To solve the quadratic equation \(x^2 + 7x + 10 = 0\), we look for two numbers that multiply to the constant term (\(10\)) and add to the coefficient of the linear term (\(7\)). These two numbers are \(2\) and \(5\), because:
\[2 \times 5 = 10\]
\[2 + 5 = 7\]
Thus, we can factor the quadratic expression as:
\[x^2 + 7x + 10 = (x + 2)(x + 5) = 0\]
According to the Zero Product Property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore:
\[x + 2 = 0 \implies x = -2\]
\[x + 5 = 0 \implies x = -5\]
These are the solutions for \(x\). The corresponding x-intercepts are the points on the graph where these values of \(x\) occur with \(y=0\), yielding the points \((-2, 0)\) and \((-5, 0)\).
Comparing our findings \((-2, 0)\) and \((-5, 0)\) to the given options:
◯ A. \((2, 0)\) and \((-5, 0)\)
◯ B. \((2, 0)\) and \((5, 0)\)
◯ C. \((-2, 0)\) and \((-5, 0)\)
◯ D. \((-2, 0)\) and \((5, 0)\)
The correct option is C. The x-intercepts of the graph of \(x^2 + 7x + 10 = 0\) are \((-2, 0)\) and \((-5, 0)\), so \(\boxed{C}\).
In geometry and physics, finding the x-intercepts of a quadratic function helps identify where a projectile, like a ball thrown in the air, hits the ground. By setting the vertical position equation to zero, we can find the exact moments the object lands, providing essential data for predicting trajectories and understanding motion in everyday situations.