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A quadratic equation is an expression of the form \( ax^2 + bx + c = 0 \). This equation is called quadratic because of the term \( x^2 \), which is the highest power of the variable \( x \).

Quadratic equations are ubiquitous in algebra and arise in various fields, such as physics, engineering, and economics. Understanding quadratic equations is crucial as they help in modeling many real-world situations.

• The term \( a \) is known as the coefficient of \( x^2 \).
• The term \( b \) is the coefficient of \( x \), and \( c \) is the constant term.

One of the key characteristics of quadratic equations is their graph, a parabola, which can open upwards or downwards depending on the sign of \( a \). In the problem provided, the equation \( y = 4 - x - x^2 \) can be rearranged to \( -x^2 - x + 4 \), which is a quadratic in disguise ready to be analyzed for intercepts.

The y-intercept of a graph is the point where the curve, which is the line or curve described by the equation, crosses the y-axis. This occurs when \( x = 0 \). It helps in providing a starting point for understanding the behavior of the equation visually when plotted.

To find the y-intercept of the equation \( y = 4 - x - x^2 \), substitute \( x = 0 \) and solve for \( y \):

• Setting \( x = 0 \) transforms the equation to \( y = 4 - 0 - 0^2 \).
• Thus, \( y = 4 \).

Hence, the y-intercept is \( (0, 4) \). This point tells us that our parabola will start at this point on the y-axis.

The x-intercepts, also known as the roots or zeros of the equation, are the points where the graph crosses the x-axis. This happens when \( y = 0 \). Finding x-intercepts is essential as it reveals the solutions to the quadratic equation.

For the quadratic equation \( 4-x-x^2 \), set \( y = 0 \):

• Rewriting gives \( 0 = -x^2 - x + 4 \).
• Reordering terms produces \( 0 = x^2 + x - 4 \), a standard quadratic form.

This equation can be solved using various methods, including factorization, completing the square, or the quadratic formula. Here, we deploy the quadratic formula to discover our x-intercepts.

The quadratic formula is a powerful tool for finding the roots of quadratic equations. It provides the solution needed, especially when the equation is not easily factorable. The formula is expressed as:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]In this context:

• \( a = 1 \)
• \( b = 1 \)
• \( c = -4 \)

This is substituted into the quadratic formula to calculate the x-intercepts:

• Calculate the discriminant: \( b^2 - 4ac = 1^2 - 4 imes 1 imes (-4) = 1 + 16 = 17 \).
• Solve for \( x \): \( x = \frac{-1 \pm \sqrt{17}}{2} \).
• This simplifies to x-intercepts of \( x = 2 \) and \( x = -2 \).

The quadratic formula not only finds the x-intercepts efficiently but also indicates the nature of the parabola based on the value of the discriminant.