Problem 40 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the \(x\) - and \(y\) -intercepts of the graph of the circle. $$x^{2}+8 x+y^{2}+2 y+9=0$$

Step-by-Step Solution

Answer

The x and y intercepts are the solutions of the quadratic equations formed in Steps 1 and 3. Exact values need to be computed.

Step 1: Finding the x-intercepts

The x-intercepts are obtained when \(y=0\). Therefore, set \(y=0\) in the given equation \[x^{2}+8 x+y^{2}+2 y+9=0\]. The simplified equation is \[x^{2}+8 x+9=0\]. This is a quadratic equation in the form \(ax^{2}+bx+c=0\) which can be solved by using the formula \(x=frac{-b ± sqrt{b^{2}-4ac}}{2a}\].

Step 2: Calculation of x-intercepts

Now, solving the equation formed in the previous step by substituting \(a=1\), \(b=8\) and \(c=9\) in the quadratic formula, two potential x-intercepts are found, \(x_{1}\) and \(x_{2}\). The exact values depend on whether the term under the square root (the discriminant) is positive or negative. If it is positive, there are two different x-intercepts; if it is negative, there are no x-intercepts; if it is zero, there is one x-intercept.

Step 3: Finding the y-intercepts

The y-intercepts are obtained when \(x=0\). Therefore, set \(x=0\) in the given equation \[x^{2}+8 x+y^{2}+2 y+9=0\]. The simplified equation is \[y^{2}+2 y+9=0\]. This equation is similar to the x-intercept equation and can be solved in a similar manner by applying the quadratic formula here as well.

Step 4: Calculation of y-intercepts

Now, solving the equation formed in the previous step by substituting \(a=1\), \(b=2\) and \(c=9\) in the quadratic formula, two potential y-intercepts are found, \(y_{1}\) and \(y_{2}\). Again, the number of intercepts depends on the value of the discriminant.

Key Concepts

Understanding X-Intercepts

To find the x-intercepts of a circle's equation, we first set the variable \(y\) to zero. Why? Because x-intercepts are where the graph touches or crosses the x-axis, and at these points, the y-value is zero. By substituting \(y = 0\) into the equation \(x^2 + 8x + y^2 + 2y + 9 = 0\), we can simplify it to just involve \(x\): \(x^2 + 8x + 9 = 0\). This is now an easier quadratic equation focused solely on \(x\).

Solving this equation using the quadratic formula will give us potential x-intercepts. The quadratic formula helps find the roots of any quadratic equation, which in this context, provides the x-values where the circle intercepts the x-axis.

Unlocking Y-Intercepts

Y-intercepts are found when \(x = 0\), because they occur where the graph meets the y-axis. At any y-intercept, the x-value is consequently zero. So, by substituting \(x = 0\) into the original circle equation, we get \(y^2 + 2y + 9 = 0\). This is another quadratic equation, this time focused on \(y\).

By using the quadratic formula again, we can solve for \(y\) to determine if and where the circle intersects the y-axis. Checking the solutions will tell you how the circle aligns horizontally with the y-axis.

Quadratic Formula Essentials

The quadratic formula is a powerful tool for finding the solutions of quadratic equations. In its universal form, it is written as:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula helps you solve equations of the form \(ax^2 + bx + c = 0\). By identifying the coefficients \(a\), \(b\), and \(c\) from our quadratic equation, we can plug them into the formula to solve for \(x\).

The symbol \(\pm\) indicates that there could be two possible solutions: one for each sign. This means that for a quadratic equation, there can potentially be two intercepts. The discovery of these intercepts is essential for understanding where a circle graph meets the axes.

Investigating the Discriminant

The discriminant is a part of the quadratic formula, expressed as \(b^2 - 4ac\). It plays a crucial role in determining the nature of the solutions for a quadratic equation. Here's how it works:

• If the discriminant is positive, \(b^2 - 4ac > 0\): There are two distinct real solutions or intercepts.
• If the discriminant is zero, \(b^2 - 4ac = 0\): There is exactly one real solution, meaning the circle just touches the axis.
• If the discriminant is negative, \(b^2 - 4ac < 0\): There are no real solutions or intercepts as the circle doesn't intersect the axis at all.

Thus, the discriminant significantly influences whether the circle will cross or just touch the axis, which is essential for plotting graphs.