If 3x² + x = 2, then x
Understand the Problem
The question is asking to solve the quadratic equation 3x² + x = 2 for the variable x. This involves rearranging the equation and applying methods for solving quadratics.
Answer
The solutions are \(x = \frac{2}{3}\) and \(x = -1\).
Answer for screen readers
The solutions to the equation (3x² + x - 2 = 0) are (x = \frac{2}{3}) and (x = -1).
Steps to Solve
- Rearrange the equation
To solve the equation (3x² + x = 2), we first need to rearrange it into standard quadratic form (ax² + bx + c = 0).
Subtract 2 from both sides to get: $$ 3x² + x - 2 = 0 $$
- Identify coefficients
In standard quadratic form (ax² + bx + c = 0), identify the coefficients:
- (a = 3)
- (b = 1)
- (c = -2)
- Apply the quadratic formula
The quadratic formula is: $$ x = \frac{-b \pm \sqrt{b² - 4ac}}{2a} $$
Substituting the values of (a), (b), and (c): $$ x = \frac{-1 \pm \sqrt{(1)² - 4(3)(-2)}}{2(3)} $$
- Calculate the discriminant
Now calculate the value inside the square root (the discriminant): $$ 1² - 4(3)(-2) = 1 + 24 = 25 $$
- Calculate the roots
Now substitute the discriminant back into the quadratic formula: $$ x = \frac{-1 \pm \sqrt{25}}{6} $$
Calculate the square root: $$ x = \frac{-1 \pm 5}{6} $$
This gives us two potential solutions for (x): 1. $$ x_1 = \frac{-1 + 5}{6} = \frac{4}{6} = \frac{2}{3} $$
$$ x_2 = \frac{-1 - 5}{6} = \frac{-6}{6} = -1 $$
The solutions to the equation (3x² + x - 2 = 0) are (x = \frac{2}{3}) and (x = -1).
More Information
The quadratic equation can have two solutions because it is a parabolic function, and these roots are where the graph meets the x-axis. If the discriminant (the value under the square root) is positive, there are two real solutions.
Tips
- Miscalculating the discriminant, which can lead to incorrect roots.
- Forgetting to rearrange the quadratic equation into standard form before applying the formula. Always check to ensure it is correctly set up as (ax² + bx + c = 0).
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