Question

We consider the plane curves $$ \begin{aligned} &X=\left\\{(a, b) \in \mathbb{R}^{2}: b-a^{3}+7 a-5=0\right\\} \\ &Y=\left\\{(a, b) \in \mathbb{R}^{2}: 20 a^{2}-5 a b-4 b^{2}+35 a+35 b-21=0\right\\} \end{aligned} $$

Short answer

Solve for points \((a, b)\) by substituting \(b\) from curve \(X\) into \(Y\) and simplifying.

Step-by-step solution

Identify Curves X and Y

The curve \(X\) is given by the equation \(b - a^3 + 7a - 5 = 0\), which can be rewritten as \(b = a^3 - 7a + 5\). This represents a cubic function in the plane. The curve \(Y\) is given by the equation \(20a^2 - 5ab - 4b^2 + 35a + 35b - 21 = 0\).

Check for Points of Intersection

To find points where both curves intersect, substitute \(b = a^3 - 7a + 5\) from curve \(X\) into the equation of curve \(Y\). This leads to the equation \(20a^2 - 5a(a^3 - 7a + 5) - 4(a^3 - 7a + 5)^2 + 35a + 35(a^3 - 7a + 5) - 21 = 0\).

Simplify and Evaluate

Simplify the substituted equation to solve for \(a\). This involves expanding the powers, combining and simplifying terms to form a solvable polynomial equation. Once \(a\) values are found, substitute back into \(b = a^3 - 7a + 5\) to find corresponding \(b\) values.

Finding Concrete Solutions

Given the complexity of the polynomial, it may require numeric or algebraic methods (e.g. factorization, synthetic division, or computational algorithms) to solve for real values of \(a\). Once these are identified, use them to calculate exact \(b\) values for those intersections.

Key concepts

Intersection of Curves

Finding where two curves intersect is like discovering where two roads cross. This is important in mathematics because intersections show common solutions or shared points between equations. When you have two equations, you can solve them simultaneously to find the intersection points. To determine where two curves intersect:

• Express one of the equations in terms of a variable, typically by isolating one variable on one side. For example, if you have a curve defined as \( b = a^3 - 7a + 5 \), it becomes easier to substitute this expression in the other equation.
• Substitute the expression from one equation into the other. This transforms one of the equations into a single-variable equation.
• Solve this new equation to find the variable's values, which gives you the intersection coordinates when substituted back into the original equations.

Once you've found the solutions, these points tell us the shared solutions of the original equations. In real-life scenarios, recognizing intersections can be key in solving complex problems, such as optimizing routes or determining shared resources.

Cubic Functions

Cubic functions are equations that involve variables raised to the third power. They form curves that can take on a variety of shapes, fitting many real-world applications.These functions are generally written as:\[ f(x) = ax^3 + bx^2 + cx + d \]where \( a \), \( b \), \( c \), and \( d \) are constants, and \( a eq 0 \).Characteristics of cubic functions:

• They can have one, two, or three real roots, meaning places where the graph touches or crosses the horizontal axis.
• The shape of the curve can vary greatly, often with a natural curve that increases or decreases sharply before leveling out.
• Cubic curves can have either one turning point or two turning points, providing local maxima or minima.

In the context of intersecting curves, identifying one curve as a cubic function, like \( b = a^3 - 7a + 5 \), can simplify finding intersection points. The polynomial nature of cubic functions also implies that solving such equations typically requires either algebraic manipulation or numerical methods.

Polynomial Solutions

Working with polynomial solutions involves identifying the roots or solutions of polynomial equations. These roots are the values of the variable that satisfy the equation. Polynomial equations can range from simple linear equations to complex cubic or even higher degree equations. Steps to finding polynomial solutions:

• Simplify the equation if possible by collecting like terms and reducing it to its simplest form.
• Factorize the polynomial if it can be expressed as a product of simpler polynomials, providing an easier path to discovering the roots.
• For higher degree polynomials, techniques such as synthetic division or computer algorithms might be necessary to find exact or approximate solutions.

When dealing with intersections of curves, once you've substituted and transformed the equations, finding the polynomial solutions helps identify the specific values of variables that result in common intersection points. Understanding polynomial behavior and manipulation is key in unraveling complex mathematical problems involving multiple equations.