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In a quadratic function such as \(y = x^2 + 4x + 4\), finding the x-intercepts involves determining the points where the graph meets the x-axis. At these points, the value of \(y\) is zero. So, we set \(y = 0\) and solve for \(x\). When we do this, the equation becomes \(0 = x^2 + 4x + 4\). To find the x-values at which the graph intersects the x-axis, we must solve this equation.
This equation is a perfect square trinomial, which can be factored into \((x + 2)^2 = 0\). The solution to this equation is \(x = -2\). Since the roots \(x = -2\) repeat, there's a single x-intercept at \(-2, 0\). This means the vertex of the parabola touches the x-axis at this point without actually crossing it, which we often describe as a "double root."
Thus,

• The x-intercept or root is \(x = -2\).

The y-intercept of a function is the point where it intersects the y-axis. This occurs when \(x = 0\) because the y-axis is at \(x = 0\). For the quadratic equation \(y = x^2 + 4x + 4\), we can find this intercept by simply substituting \(x = 0\) into the equation.
Substituting \(x = 0\) gives:\[ y = (0)^2 + 4(0) + 4 = 4 \] Thus, when \(x = 0\), \(y = 4\). This makes the y-intercept the point \((0, 4)\). So, the graph of this quadratic equation intersects the y-axis at the point located four units up from the origin.
To summarize,

• The y-intercept is \((0, 4)\).

Solving quadratic equations is a fundamental skill in algebra, often requiring us to find the points where the graph touches, crosses, or interacts with the axes. In general, a quadratic equation in the standard form is \(ax^2 + bx + c = 0\). There are several methods to solve these, including factoring, using the quadratic formula, or completing the square.
For our specific quadratic function \(y = x^2 + 4x + 4\), we used factoring to find the x-intercepts. Since it factored neatly into \((x + 2)^2 = 0\), it illustrated how sometimes the equation can be simplified to reveal "double roots," where the parabola merely touches the x-axis at a single point.
Use these tips to solve quadratic equations effectively:

• Check if it can be factored easily.
• If not, the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) is another reliable method.
• Always double-check by substituting the roots back into the original equation.

Knowing how to solve these equations helps you understand the behavior of parabolic graphs and find critical points such as intercepts.