Q.Find the values of $x$ for which the functions $f(x) = 3x^2 - 1$ and $g(x) = 3 + x$ are equal

To find the values of $x$ for which two functions are equal, we set their expressions equal to each other. This forms an equation that can be solved for $x$.

We are given $f(x) = 3x^2 - 1$ and $g(x) = 3 + x$.

1. Set $f(x) = g(x)$: $3x^2 - 1 = 3 + x$
2. Rearrange the equation into the standard quadratic form $ax^2 + bx + c = 0$: $3x^2 - x - 1 - 3 = 0$ $3x^2 - x - 4 = 0$
3. Solve the quadratic equation by factorization: $3x^2 - 4x + 3x - 4 = 0$ $x(3x - 4) + 1(3x - 4) = 0$ …

When we are asked to find the values of $x$ for which two functions, say $f(x)$ and $g(x)$, are equal, we are essentially looking for the input values of $x$ where both functions produce the exact same output value. Graphically, this corresponds to finding the $x$-coordinates of the points where the graphs of $y = f(x)$ and $y = g(x)$ intersect.

To solve this algebraically, we simply set the expressions for $f(x)$ and $g(x)$ equal to each other and then solve the resulting equation for $x$.

1. Set the function expressions equal.

   The problem states that $f(x)$ and $g(x)$ are equal. So, we write:

   $f(x) = g(x)$

2. Substitute the given function definitions.

   We are given $f(x) = 3x^2 - 1$ and $g(x) = 3 + x$. Substituting these into the equation from Step 1:

   $3x^2 - 1 = 3 + x$

3. Rearrange the equation into a standard quadratic form.

   To solve this equation, we need to bring all terms to one side, setting the equation to zero. This will result in a standard quadratic equation of the form $ax^2 + bx + c = 0$.

   First, subtract $x$ from both sides:

   $3x^2 - x - 1 = 3$

   Next, subtract $3$ from both sides:

   $3x^2 - x - 1 - 3 = 0$

   This simplifies to:

   $3x^2 - x - 4 = 0$

   A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where $a, b, c$ are constants and $a \neq 0$.

4. Solve the quadratic equation.

   We now have the quadratic equation $3x^2 - x - 4 = 0$. We can solve this using either factorization or the quadratic formula.

   Method 1: Factorization

   We look for two numbers that multiply to $a \cdot c = 3 \cdot (-4) = -12$ and add up to $b = -1$. These numbers are $-4$ and $3$.

   We can rewrite the middle term, $-x$, as $-4x + 3x$:

   $3x^2 - 4x + 3x - 4 = 0$

   Now, factor by grouping:

   $x(3x - 4) + 1(3x - 4) = 0$

   $(3x - 4)(x + 1) = 0$

   For the product of two factors to be zero, at least one of the factors must be zero: …