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A quadratic equation is any equation that can be rearranged to the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). These equations are so named because their highest exponent is 2, making them second-degree polynomials.
Quadratic equations are typically represented as parabolas when graphed. The graph can open upwards or downwards, depending on the sign of \( a \). An important property of these parabolas is that they are symmetric about a vertical line called the axis of symmetry.
Quadratics often appear in real-life scenarios, such as calculating projectile motion or finding maximum areas. Solving them involves finding the values of \( x \) that satisfy the equation. One common method is factoring, just as we did in the original exercise. Other methods include using the quadratic formula and completing the square.

• Parabola shape: The graph forms a parabola.
• Vertex: The highest or lowest point is called the vertex.
• Axis of symmetry: A vertical line passing through the vertex.

The x-intercept(s) of a graph are the point(s) where the graph crosses the x-axis. This occurs when the y-value of the function is zero. For quadratic equations, there can be one x-intercept if the vertex lies on the x-axis, two x-intercepts if the parabola crosses the axis twice, or none if it does not cross the x-axis at all.
Finding the x-intercepts requires setting the equation equal to zero and solving for \( x \). In the equation \( y = x^2 - 9 \), setting \( y = 0 \) gives \( x^2 - 9 = 0 \). We then factor the quadratic: \( (x-3)(x+3) = 0 \). This results in two solutions, \( x = 3 \) and \( x = -3 \). Thus, these points are the x-intercepts.

• Setting \( y = 0 \): To find x-intercepts, set the equation to zero.
• Factoring: Break down the quadratic into two factors.
• Solutions: The x-values from the factored form.

The y-intercept is the point where a graph intersects the y-axis. This occurs when \( x = 0 \), as any other point on the y-axis has an x-value of zero. Finding the y-intercept involves plugging \( x = 0 \) into the equation to solve for \( y \).
In our given quadratic equation \( y = x^2 - 9 \), setting \( x = 0 \) results in \( y = 0^2 - 9 \), simplifying to \( y = -9 \). Therefore, the y-intercept of the graph is at the point \( (0, -9) \). This is where the parabola crosses the y-axis.

• Setting \( x = 0 \): To find the y-intercept, the value of \( x \) is set to zero.
• Plug into equation: Substitute zero for \( x \) in the equation.
• Result: The y-value obtained is the y-intercept.