Complete Step-by-Step Solution Process for ax² + b = c
Solving quadratic equations of the form ax² + b = c follows a three-step algebraic process taught in Grade 11 enVision Algebra 1 (Chapter 9: Solving Quadratic Equations): first subtract b from both sides to isolate the ax² term, then divide both sides by a to isolate x², and finally take the square root of both sides remembering to include both positive and negative roots (x = ±√((c−b)/a)). The ± sign is critical — omitting it loses one solution. The square root step only applies when (c−b)/a ≥ 0.
Key Concepts
To solve equations of the form $ax² + b = c$, follow these steps: 1. Subtract $b$ from both sides: $ax² = c b$ 2. Divide both sides by $a$: $x² = \frac{c b}{a}$ 3. Take the square root of both sides: $x = ±\sqrt{\frac{c b}{a}}$ (when $\frac{c b}{a} \geq 0$).
Common Questions
What are the three steps to solve ax² + b = c?
Step 1: Subtract b from both sides to get ax² = c − b. Step 2: Divide by a to get x² = (c − b)/a. Step 3: Take the square root of both sides: x = ±√((c − b)/a).
Why do you include ± when taking the square root?
Because both the positive and negative square root satisfy x² = k. Omitting ± means missing one solution.
What if (c − b)/a is negative?
If (c − b)/a < 0, there are no real solutions — you cannot take the square root of a negative number in the real number system.
What if (c − b)/a equals zero?
There is exactly one solution: x = 0. Both the positive and negative root are zero, so there is only one distinct answer.
Solve 3x² + 2 = 14 step by step.
Step 1: 3x² = 12. Step 2: x² = 4. Step 3: x = ±√4 = ±2. The solutions are x = 2 and x = −2.
What algebraic property justifies dividing both sides by a?
The division property of equality: dividing both sides of an equation by the same non-zero number preserves equality.