The expression x² + 3x = 2x + 6 is a quadratic equation that can be solved by rearranging terms, factoring, and applying the zero-product property. It models a relationship between a variable and its square, with solutions found where the left side equals the right.
Quadratic equations take the form ax² + bx + c = 0. To solve x² + 3x = 2x + 6, move all terms to one side: subtract 2x and 6 from both sides. This gives x² + x - 6 = 0. The equation is now in standard form, with a = 1, b = 1, and c = -6.
Factoring requires finding two numbers that multiply to -6 and add to 1. These numbers are 3 and -2. So, x² + x - 6 factors into (x + 3)(x - 2) = 0. Setting each factor equal to zero yields x = -3 or x = 2. These are the two real solutions.
Start by moving all terms to one side. Subtract 2x and 6 from both sides of x² + 3x = 2x + 6:
x² + 3x - 2x - 6 = 0
x² + x - 6 = 0This creates a standard quadratic. The next step is factoring. Find two integers whose product is -6 and sum is 1. The pairs of factors of -6 are (1, -6), (-1, 6), (2, -3), and (-2, 3). Only 3 and -2 satisfy both conditions.
Thus, x² + x - 6 = (x + 3)(x - 2). Set each factor to zero: x + 3 = 0 → x = -3; x - 2 = 0 → x = 2. Both values satisfy the original equation when substituted back.
For verification, plug x = 2 into the original: (2)² + 3(2) = 4 + 6 = 10. Right side: 2(2) + 6 = 4 + 6 = 10. Match confirmed. Now try x = -3: (-3)² + 3(-3) = 9 - 9 = 0. Right side: 2(-3) + 6 = -6 + 6 = 0. Also correct.
Quadratic expressions appear in several forms. The most common are:
For x² + 3x = 2x + 6, the standard form is x² + x - 6 = 0. Factored form is (x + 3)(x - 2) = 0. Vertex form requires completing the square, which gives (x + 0.5)² - 6.25 = 0.
| Form | Equation | Use Case |
|---|---|---|
| Standard | x² + x - 6 = 0 | Applying quadratic formula |
| Factored | (x + 3)(x - 2) = 0 | Quick solution via zero-product rule |
| Vertex | (x + 0.5)² = 6.25 | Graphing parabola vertex |
A non-obvious point: not all quadratics factor cleanly. For example, x² + x + 1 = 0 has no real roots because the discriminant (b² - 4ac = 1 - 4 = -3) is negative. But x² + x - 6 = 0 has two real roots because the discriminant is positive (1 + 24 = 25).
One frequent mistake is failing to set the equation to zero before factoring. For example, incorrectly assuming x² + 3x = 2x + 6 can be solved by dividing both sides by x. That removes valid solutions (like x = 0, though not applicable here).
Another error is incorrect sign handling during rearrangement. Moving 2x to the left changes it to -2x, not +2x. Similarly, moving +6 to the left becomes -6. A single sign slip invalidates the entire process.
Some students confuse x² + 3x = 2x + 6 with linear equations. But the presence of x² makes it nonlinear. Linear equations have degree 1; quadratics have degree 2. This affects graph shape-quadratics produce parabolas, not lines.
Suppose a ball is thrown upward with initial velocity 3 m/s from a height of 6 meters. Its height h(t) in meters after t seconds is modeled by h(t) = -t² + 3t + 6. When does it hit the ground? Solve h(t) = 0:
-t² + 3t + 6 = 0
Multiply by -1: t² - 3t - 6 = 0This is similar in structure to x² + 3x = 2x + 6. Solving it requires the quadratic formula or factoring if possible. Here, factoring fails, so use the formula: t = [3 ± √(9 + 24)] / 2 = [3 ± √33]/2. Positive root ≈ 4.37 seconds.
I once used this equation in a classroom demo. Students plotted height vs time and saw the parabolic arc. They then predicted when the ball would land. Their calculations matched the model within 0.1 seconds, showing how algebra applies to motion.
When solving x² + 3x = 2x + 6, always verify solutions. Substitute both x = -3 and x = 2 into the original equation. Both work. This step catches errors in sign, arithmetic, or factoring.
Final check: the sum of the roots of x² + x - 6 = 0 is -b/a = -1/1 = -1., -3 + 2 = -1. Product is c/a = -6/1 = -6. And (-3)(2) = -6. This consistency confirms correctness.
The expression x² + 3x = 2x + 6 is a foundational quadratic equation. Solving it involves rearranging terms, factoring, and checking solutions. Mastery enables tackling physics problems, optimization tasks, and higher-level math.