Solve for a & b: Real Number Pairs $(a,\,b)$

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In summary, solving for a and b in the context of real number pairs means finding the values of a and b that satisfy a given equation or system of equations. Real number pairs are simply two numbers from the set of real numbers, including positive, negative, or zero numbers. Both linear and quadratic equations, as well as systems of equations, can be solved for a and b. Solving for a and b is important because it allows us to find specific solutions to equations and systems of equations, which can be useful in various fields. Different methods, such as substitution, elimination, graphing, and the quadratic formula, can be used to solve for a and b in real number pairs.
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Determine the pair(s) of real numbers $(a,\,b)$ that satisfy the equation $16a^2b^2-48a^2b+24ab^2+100a^2+16b^2-72ab+150a-48b+100=28$.
 
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  • #2
Rewrite the equation as:
$$16b^2(a^2+1)-48b(a^2+1)+100(a^2+1)+24ab^2-72ab+150a=28$$
$$\Rightarrow 4(a^2+1)(4b^2-12b+25)+6a(4b^2-12b+25)=28$$
$$\Rightarrow (2a^2+3a+2)(4b^2-12b+25)=14$$

The minimum value of $2a^2+3a+2$ is $7/8$ at $a=-3/4$ and that of $4b^2-12b+25$ is $16$ at $b=3/2$. Clearly, the pair $(a,b)$ is the $(-3/4,3/2)$.
 
  • #3
Bravo, Pranav!(Clapping) And thanks for participating!:)
 

FAQ: Solve for a & b: Real Number Pairs $(a,\,b)$

1. What does it mean to "solve for a & b" in the context of real number pairs?

Solving for a and b in this context means finding the values of a and b that satisfy a given equation or system of equations. In other words, it is finding the specific pair of real numbers (a, b) that make the equation or system of equations true.

2. What are real number pairs?

Real number pairs are simply two numbers, a and b, that belong to the set of real numbers. They can be positive, negative, or zero, and can include fractions, decimals, and irrational numbers.

3. What types of equations or systems of equations can be solved for a & b?

In the context of real number pairs, a & b can be solved for in any equation or system of equations that involves two unknowns (a and b) and two equations. This could include linear equations, quadratic equations, or systems of equations that combine both types.

4. Why is solving for a & b important?

Solving for a & b allows us to find specific solutions to equations or systems of equations. This can be useful in a variety of fields, including mathematics, physics, and engineering, as it allows us to find the values of variables that satisfy a given set of conditions.

5. What methods can be used to solve for a & b?

There are several methods that can be used to solve for a & b in real number pairs. These include substitution, elimination, graphing, and the quadratic formula. The specific method used will depend on the type of equation or system of equations being solved.

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