Use factoring to solve the polynomial equation:
$2 x^2=2-3 x$
To solve the equation \(2x^2 = 2 - 3x\), we first write it in the standard quadratic form \(ax^2 + bx + c = 0\). By moving all terms to the left side of the equation, we obtain:
\[2x^2 + 3x - 2 = 0\]
Here, the coefficients are \(a = 2\), \(b = 3\), and \(c = -2\).
We need to factor the quadratic trinomial \(2x^2 + 3x - 2\). We look for two numbers that multiply to \(a \times c = 2 \times (-2) = -4\) and add up to \(b = 3\).
The factors of \(-4\) are \((1, -4), (-1, 4), (2, -2)\).
Checking their sums:
\(1 + (-4) = -3\)
\(-1 + 4 = 3\)
The numbers we need are \(4\) and \(-1\).
Now, we rewrite the middle term \(3x\) using the two numbers found:
\[2x^2 + 4x - x - 2 = 0\]
Next, we factor by grouping:
\[(2x^2 + 4x) - (x + 2) = 0\]
\[2x(x + 2) - 1(x + 2) = 0\]
Factoring out the common binomial \((x + 2)\), we get:
\[(2x - 1)(x + 2) = 0\]
To find the roots, we set each factor equal to zero:
Thus, the solutions are \(x = \frac{1}{2}\) or \(x = -2\).
Factoring quadratic equations is essential for solving real-world problems involving motion, such as determining when an object thrown into the air will reach a specific height. For instance, if the height of a projectile is modeled by a quadratic equation, setting the height to a constant and rearranging the terms allows us to use factoring to find the exact time(s) the object is at that height. This foundational skill helps in physics, engineering, and economics for optimizing processes or predicting outcomes.