Use factoring to solve the polynomial equation:
$2 x^2 = 2 - 3 x$
The goal is to solve the quadratic equation \(2x^2 = 2 - 3x\) by factoring. First, we must write the equation in the standard form \(Ax^2 + Bx + C = 0\). We can do this by adding \(3x\) and subtracting \(2\) from both sides:
\[2x^2 + 3x - 2 = 0\]
Here, the coefficients are \(A = 2\), \(B = 3\), and \(C = -2\).
To factor the quadratic expression \(2x^2 + 3x - 2\), we look for two numbers that multiply to the product of \(A\) and \(C\) (which is \(2 \times (-2) = -4\)) and add up to \(B\) (which is \(3\)).
Testing the factors of \(-4\):
So, we can split the middle term \(3x\) as \(4x - x\).
Now, rewrite the quadratic equation using the split middle term:
\[2x^2 + 4x - x - 2 = 0\]
We group the terms to facilitate factoring:
\[(2x^2 + 4x) - (x + 2) = 0\]
Factor out the common terms from each group:
\[2x(x + 2) - 1(x + 2) = 0\]
Now, factor out the common binomial \((x + 2)\):
\[(2x - 1)(x + 2) = 0\]
Set each factor to zero to solve for \(x\):
1) \(2x - 1 = 0 \implies 2x = 1 \implies x = \frac{1}{2}\)
2) \(x + 2 = 0 \implies x = -2\)
Thus, the solutions are \(x = \frac{1}{2}\) or \(x = -2\). The final answer is \(\boxed{x = \frac{1}{2}, x = -2}\).
Quadratic equations are widely used in physics and engineering. For example, if you launch a projectile, the height \(h\) as a function of time \(t\) is often modeled by a quadratic equation like \(h(t) = -16t^2 + v_0t + h_0\). Solving such an equation by factoring or using the quadratic formula allows us to determine the exact time when the object hits the ground (when \(h(t) = 0\)). This helps in predicting trajectories, analyzing motion, and designing structures safely.