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The x-intercepts of a quadratic function are the points where the graph of the function crosses the x-axis. At the x-intercepts, the y value is equal to zero. To find the x-intercepts of a quadratic equation like \( y = x^2 - 5x + 6 \), you need to set \( y \) equal to zero and solve the resulting equation \( 0 = x^2 - 5x + 6 \).

There are several methods to solve this equation for the x-intercepts, including:

• **Factoring** - Splitting the equation into products of simpler binomials if possible.
• **Quadratic Formula** - Using the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the roots.
• **Completing the Square** - Rewriting the equation in the form of a perfect square trinomial.

For the equation \( 0 = x^2 - 5x + 6 \), factoring is straightforward: it factors into \( 0 = (x - 2)(x - 3) \). This tells us that if \( (x - 2) = 0 \) or \( (x - 3) = 0 \), the product will be zero. Thus, \( x = 2 \) or \( x = 3 \), marking the x-intercepts at points \((2, 0)\) and \((3, 0)\).

The y-intercepts of a quadratic function give us the points where the graph touches or crosses the y-axis. This occurs when the x-value is zero. Finding the y-intercept involves substituting \( x = 0 \) into the equation and solving for \( y \).

For the function \( y = x^2 - 5x + 6 \), if we set \( x = 0 \), the equation simplifies to \( y = (0)^2 - 5(0) + 6 \). This computation results in \( y = 6 \).

Hence, the y-intercept of the function is at the point \((0, 6)\). This means when the input, or the x-value, is zero, the output, or y-value, is 6. The point \((0, 6)\) visually represents where the quadratic graph crosses or touches the y-axis.

The factoring method is a common technique used to solve quadratic equations, especially when the equation is easily factorable. Quadratics are often expressed in the form \( ax^2 + bx + c = 0 \).

The goal in factoring is to express the quadratic as a product of simpler binomials, \((mx + n)(px + q) = 0\). If a quadratic can be factored, the roots or solutions of the quadratic equation are where the product of the binomials equals zero. This is based on the zero product property, which states that if the product of two expressions is zero, at least one of the expressions must be zero.

For the example \( x^2 - 5x + 6 = 0 \), it can be factored into \((x - 2)(x - 3) = 0\). This indicates that either \(x - 2 = 0\) or \(x - 3 = 0\), resulting in \(x = 2\) or \(x = 3\).

Factoring leverages the simplicity of these factored components, making it a fast and efficient approach when applicable. However, if a quadratic cannot be factored neatly, other methods like the quadratic formula might be necessary.