We are given two equations:

1. $a + b = 31$
2. $a \times b = 234$

Let's solve for $a$ and $b$.

Step 1: Express one variable in terms of the other

From the first equation: $a = 31 - b$

Step 2: Substitute into the second equation

Substitute $a = 31 - b$ into the second equation $a \times b = 234$: $(31 - b) \times b = 234$

Step 3: Expand and simplify the equation

$31b - b^2 = 234$ Rearrange the terms to form a quadratic equation: $b^2 - 31b + 234 = 0$

Step 4: Solve the quadratic equation

We can solve this quadratic equation using the quadratic formula: $b = \frac{-(-31) \pm \sqrt{(-31)^2 - 4(1)(234)}}{2(1)}$ $b = \frac{31 \pm \sqrt{961 - 936}}{2}$ $b = \frac{31 \pm \sqrt{25}}{2}$ $b = \frac{31 \pm 5}{2}$

Step 5: Find the two possible values of $b$

1. $b = \frac{31 + 5}{2} = \frac{36}{2} = 18$
2. $b = \frac{31 - 5}{2} = \frac{26}{2} = 13$

Step 6: Find the corresponding values of $a$

If $b = 18$, then $a = 31 - 18 = 13$. If $b = 13$, then $a = 31 - 13 = 18$.

Final Answer:

The two possible solutions are:

1. $a = 13$, $b = 18$
2. $a = 18$, $b = 13$

Do you want further details or have any questions?

Here are 5 related questions:

1. How do you solve quadratic equations by factoring?
2. Can you explain the discriminant in the quadratic formula?
3. How can simultaneous equations be solved using substitution?
4. What are the different methods for solving a system of equations?
5. How can systems of nonlinear equations be approached?

Tip: When solving simultaneous equations, substitution is a powerful method if one equation is easily expressed in terms of one variable.