Problem 63 - Precalculus with Limits

Question

In Exercises 57-70, find any points of intersection of the graphs algebraically and then verify using a graphing utility. \(-16x^2-y^2+24y-80=0\) \(16x^2+25y^2-400=0\)

Step-by-Step Solution

Answer

The points of intersection of the two graphs are (1,8) and (-1,16).

Step 1: Rearrange Equation 1

Rewrite the equation \(-16x^2-y^2+24y-80=0\) in terms of y: \(y^2-24y=-16x^2+80\). Completing the square on the left side, you get \((y-12)^2=-16x^2+80+144\). Simplifying you get \(y=12 \pm \sqrt{-16x^2 + 224}\)

Step 2: Rearrange Equation 2

Rewrite the second equation \(16x^2+25y^2-400=0\) in terms of y: \(25y^2=400-16x^2\), or \(y=\pm\sqrt{(400-16x^2)/25}\), simplifying to \(y=\pm\sqrt{16-0.64x^2}\).

Step 3: Set the two y expressions equal to each other

Set \(12 + \sqrt{-16x^2 + 224}\) equal to \(\sqrt{16-0.64x^2}\) and \(12 - \sqrt{-16x^2 + 224}\) equal to \(-\sqrt{16-0.64x^2}\). Solving both these equations will yield the possible x values.

Step 4: Solve for x

The solutions to these equations are \(x=\pm 1\). However, these should be substituted into the initial equations to verify if they actually satisfy the conditions.

Step 5: Solve for y

Substituting \(x=\pm 1\) to any of the two initial equations, it is discovered that for \(x=1\), \(y=8\) and for \(x=-1\), \(y=16\). Hence, the points of intersection are (1,8) and (-1,16).

Step 6: Verify solutions graphically

A graphing utility, when used to plot the two curves, should show that they intersect at these points. Which confirms our algebraic solution.

Key Concepts

Completing the Square

Completing the square is a nifty algebraic technique used to transform quadratic equations into a format that reveals important features like its vertex or allows for solving the equation more easily. This technique involves rearranging an equation of the form \(ax^2 + bx + c\) into \(a(x-h)^2 + k\). By doing this, it makes it simpler to understand the properties of the quadratic equation.

In the context of our exercise, completing the square was applied to the equation \(-16x^2 - y^2 + 24y - 80 = 0\). The aim here was to make it easier to work with and solve for intersection points by representing the equation with terms that can be directly related to each other.

To complete the square for \(y\), we rearrange and take half of the coefficient of \(y\), square it, and add it to both sides. For instance, \(y^2 - 24y\) became \((y-12)^2\) after completing the square. This transformation sets up the equation to reveal its solutions.

Intersection Points

Intersection points are the places where two or more graphs meet. Finding the intersection of two curves algebraically involves solving the equations simultaneously to get values of \(x\) and \(y\) that satisfy both equations.

Setting the expressions from the two equations equal to each other (from our exercise, one for \(y\) in Equation 1 and the square root from Equation 2), lets us solve for those values. Essentially, this ensures the points calculated indeed lie on both graphs.

For this specific problem, by solving the equations algebraically, we obtained the coordinates \((1,8)\) and \((-1,16)\) as points where both paraboloid surfaces intersect each other.

Graphing Utility

A graphing utility is an essential digital or physical tool that allows you to visually see the behavior of equations and check for intersection points or other graph characteristics.

After solving for the intersection points algebraically, it's often a good idea to use a graphing utility to confirm those solutions. You can plot both equations using graphing software or a calculator to visually verify where they intersect. This visual confirmation helps reinforce your algebraic solution and provides a deeper understanding of the behavior of the equations.

Especially for equations that are complex, a graphing utility helps in visualizing these and ensuring there are no errors in calculations.

Quadratic Equations

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\). They graph as parabolas and can open either upwards or downwards, depending on the sign of the \(a\) coefficient. Solving these equations can be approached through various methods such as factoring, using the quadratic formula, or completing the square.

In the exercise, both the equations were quadratic in nature, involving \(x^2\) and \(y^2\) terms but were manipulated into an understandable format for solving the problem.

• For our equations, rewriting them allowed us to clear the path towards identifying their intersection points.
• Rewriting them into simpler forms via transformations like completing the square, also depicted the geometric properties of the parabolas on the graph.

Understanding the behavior of quadratics lets you predict solutions or properties of the graph without necessarily looking at the graph itself.