- Rewrite the equation by moving all terms to one side: \(16x^2 + 16x - 32 = 0\).
- Simplify the equation by dividing by the common factor of \(16\), resulting in \(x^2 + x - 2 = 0\).
- Factor the simplified quadratic expression into \((x + 2)(x - 1) = 0\).
- Identify the roots as \(x = -2\) and \(x = 1\), which corresponds to the third provided option.
- The solutions to the equation are \(\boxed{x = -2, x = 1}\).
Explanation
- Problem Analysis and Standard Form Preparation
We are given the quadratic equation \(16x^2 + 10x - 27 = -6x + 5\). Our objective is to find the values of \(x\) that satisfy this equation. We begin by organizing all the terms on one side to transform this into the standard form of a quadratic equation, \(ax^2 + bx + c = 0\).
- Simplifying the Equation
To group all terms on the left side, we add \(6x\) to both sides and subtract \(5\) from both sides of the equation:
\[16x^2 + 10x + 6x - 27 - 5 = 0\]
Combining the like terms, we get:
\[16x^2 + 16x - 32 = 0\]
Now we have a standard quadratic equation. To simplify it, we notice that each coefficient is divisible by \(16\). Dividing the entire equation by \(16\), we obtain:
\[x^2 + x - 2 = 0\]
- Factoring and Finding Roots
We now need to factor the quadratic trinomial \(x^2 + x - 2\). We look for two numbers that multiply to \(-2\) and add to \(1\). Those two numbers are \(2\) and \(-1\). Thus, we can write the factored form as:
\[(x + 2)(x - 1) = 0\]
To find the roots, we set each factor equal to zero:
\[x + 2 = 0 \quad \text{or} \quad x - 1 = 0\]
Solving these gives \(x = -2\) or \(x = 1\). Comparing these results to the given options, we find they correspond to the third option.
Examples
Quadratic equations are widely used in physics and engineering to model motion. For instance, if you throw a ball, its height \(h\) over time \(t\) can often be described by a quadratic function \(h(t) = -gt^2 + vt + h_0\). Solving for \(t\) when \(h=0\) helps determine when the ball will hit the ground. Understanding how to simplify and solve these equations is a foundational skill for predicting the behavior of moving objects.