You know, sometimes math problems feel like a secret code, don't they? We're given this expression, 3x² - x - 4, and asked to find its 'zeros' and then, importantly, to verify them. It sounds a bit technical, but at its heart, it's about finding the specific input values (the 'x's) that make the entire expression equal to zero. Think of it like finding the exact spot on a graph where a curve crosses the x-axis.
So, how do we crack this code? The fundamental idea, as we see in various mathematical explorations, is to set the expression equal to zero. Our goal is to solve for 'x'. So, we're looking at:
3x² - x - 4 = 0
Now, for a quadratic equation like this, there are a few trusty methods. We could try factoring, which is like breaking down a complex number into its simpler building blocks. Or, if factoring feels a bit elusive, the quadratic formula is always there to lend a hand. It's a reliable tool that works for any quadratic equation.
Looking at our specific equation, 3x² - x - 4 = 0, we can use the quadratic formula. This formula is derived from the general form ax² + bx + c = 0, where in our case, a = 3, b = -1, and c = -4. The formula itself is:
x = [-b ± √(b² - 4ac)] / 2a
Let's plug in our values:
x = [ -(-1) ± √((-1)² - 4 * 3 * -4) ] / (2 * 3)
x = [ 1 ± √(1 + 48) ] / 6
x = [ 1 ± √49 ] / 6
x = [ 1 ± 7 ] / 6
This gives us two potential solutions:
x = (1 + 7) / 6 = 8 / 6 = 4/3x = (1 - 7) / 6 = -6 / 6 = -1
So, we've found our potential zeros: x = 4/3 and x = -1.
But the query also asks us to verify these. This is the crucial step to ensure our calculations are spot on. Verification means plugging these 'x' values back into the original expression 3x² - x - 4 and checking if the result is indeed zero.
Let's test x = 4/3:
3 * (4/3)² - (4/3) - 4
3 * (16/9) - 4/3 - 4
16/3 - 4/3 - 12/3 (converting 4 to 12/3 for common denominator)
(16 - 4 - 12) / 3
0 / 3 = 0
Success! x = 4/3 is indeed a zero.
Now, let's test x = -1:
3 * (-1)² - (-1) - 4
3 * (1) + 1 - 4
3 + 1 - 4
4 - 4 = 0
And there we have it! x = -1 also makes the expression equal to zero.
It's always satisfying when the numbers line up perfectly. Finding the zeros of a function, like 3x² - x - 4, is a fundamental step in understanding its behavior, and the verification process is our way of double-checking our work, ensuring we've truly unlocked the code.