Which expression is equal to $5\left(2 x^2-1\right)+3\left(7 x^2+1\right)$ ?
We are given the algebraic expression \(5(2x^2 - 1) + 3(7x^2 + 1)\) and asked to simplify it by distributing and combining like terms. This involves using the distributive property, \(a(b + c) = ab + ac\), to expand the products and then aggregating the terms involving \(x^2\) and the constant terms separately.
First, we apply the distributive property to each term. For the first part, we have:
\[5(2x^2 - 1) = 5 \cdot 2x^2 - 5 \cdot 1 = 10x^2 - 5.\]
For the second part, we have:
\[3(7x^2 + 1) = 3 \cdot 7x^2 + 3 \cdot 1 = 21x^2 + 3.\]
Now, we add these two results together:
\[(10x^2 - 5) + (21x^2 + 3).\]
To combine like terms, we group the terms with \(x^2\) and the constant terms separately:
\[(10x^2 + 21x^2) + (-5 + 3).\]
Adding the coefficients of \(x^2\):
\[10 + 21 = 31.\]
Adding the constant terms:
\[-5 + 3 = -2.\]
Therefore, the simplified expression is \(31x^2 - 2\). Comparing this to the given options:
We see that the correct choice is (d).
In many engineering and physics applications, algebraic simplification is a fundamental skill. For instance, when analyzing the total energy of a system composed of different parts, you might end up with an expression like \(E = E_1(x) + E_2(x)\). By expanding and combining like terms, you can determine the overall behavior of the system, such as how total potential energy scales with a distance parameter \(x\). This helps in optimizing designs, such as finding the equilibrium position where the net force is zero.