When solving quadratic equations in the form ax² + bx + c = 0, where a is not equal to 1, there are several effective methods. The most common are factoring by grouping and using the quadratic formula.
1. Factoring by Grouping
This method involves finding two numbers that satisfy specific conditions related to the coefficients a, b, and c.
Steps:
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Find two numbers: Identify two numbers whose product is equal to ac (a multiplied by c) and whose sum is equal to b. Let's call these numbers m and n.
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Rewrite the middle term: Replace the bx term with mx + nx. The quadratic equation now becomes ax² + mx + nx + c = 0.
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Factor by grouping: Group the first two terms and the last two terms and factor out the greatest common factor (GCF) from each group. This should result in a common binomial factor.
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Factor out the common binomial: Factor out the common binomial factor, leaving you with two factors.
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Set each factor to zero and solve: Set each factor equal to zero and solve for x. These are the solutions to the quadratic equation.
Example:
Solve 2*x² + 7x + 3 = 0
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ac = 2 * 3 = 6. We need two numbers that multiply to 6 and add to 7. Those numbers are 6 and 1.
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Rewrite the equation: 2*x² + 6x + 1x + 3 = 0
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Factor by grouping:
- 2x(x + 3) + 1(x + 3) = 0
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Factor out the common binomial: (x + 3)(2*x + 1) = 0
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Set each factor to zero and solve:
- x + 3 = 0 => x = -3
- 2x + 1 = 0 => x = -1/2
Therefore, the solutions are x = -3 and x = -1/2.
2. Quadratic Formula
The quadratic formula is a universal method that works for any quadratic equation, regardless of the value of a.
Formula:
For a quadratic equation ax² + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b² - 4ac)) / (2a)
Steps:
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Identify a, b, and c: Determine the values of a, b, and c from the quadratic equation.
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Substitute into the formula: Substitute the values of a, b, and c into the quadratic formula.
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Simplify: Simplify the expression under the square root (the discriminant) and the entire formula to find the two possible values for x.
Example:
Solve 2*x² + 7x + 3 = 0
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a = 2, b = 7, c = 3
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Substitute: x = (-7 ± √(7² - 4 2 3)) / (2 * 2)
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Simplify:
- x = (-7 ± √(49 - 24)) / 4
- x = (-7 ± √25) / 4
- x = (-7 ± 5) / 4
Therefore:
- x = (-7 + 5) / 4 = -2 / 4 = -1/2
- x = (-7 - 5) / 4 = -12 / 4 = -3
The solutions are x = -3 and x = -1/2, the same results as factoring.
Other methods
Completing the square is another valid method to solve any quadratic, including those where a is not 1. However, it can become computationally intensive for some values of a, b, and c. Therefore, the first two mentioned are usually preferred.
In summary, when solving quadratic equations where a is not 1, factoring by grouping and the quadratic formula are the most reliable and commonly used methods. Choose the method that best suits your comfort level and the specific equation you are solving.
