Solve the quadratic equation:
$12x^2 + 20x + 8 = 0$
The given equation is \(12x^2 + 20x + 8 = 0\). To solve for \(x\), we first identify the coefficients \(a = 12\), \(b = 20\), and \(c = 8\). We notice that each coefficient is divisible by the greatest common divisor, which is 4. By dividing the entire equation by 4, we simplify the equation to \(3x^2 + 5x + 2 = 0\). This makes the task of finding the roots more straightforward.
To factor the quadratic expression \(3x^2 + 5x + 2\), we look for two numbers that multiply to \(3 \times 2 = 6\) and add up to the middle coefficient 5. These two numbers are 2 and 3. We rewrite the middle term \(5x\) as \(3x + 2x\):
\[3x^2 + 3x + 2x + 2 = 0.\]
Now, we can factor by grouping. Grouping the terms gives:
\[(3x^2 + 3x) + (2x + 2) = 0.\]
Factoring out common factors from each group:
\[3x(x + 1) + 2(x + 1) = 0.\]
Finally, factoring out the common binomial \((x + 1)\), we obtain:
\[(3x + 2)(x + 1) = 0.\]
For the product of two factors to be zero, at least one of the factors must be zero. Therefore, we set each factor to zero and solve for \(x\):
The roots of the equation \(12x^2 + 20x + 8 = 0\) are \(x = -1\) and \(x = -\frac{2}{3}\).
Quadratic equations like these are fundamental in physics and engineering. For example, if you are modeling the trajectory of a ball thrown into the air, the height \(h(t)\) at time \(t\) might be given by a quadratic equation. Finding the roots of the equation allows you to determine exactly when the ball hits the ground (i.e., when \(h(t) = 0\)). This ability to solve for specific time intervals is crucial for predicting motion and designing various mechanical systems.