Methods: Factoring, completing the square, quadratic formula.
for these equations:
1. x^2 - x = 6
2. -2x^2 + 5x - 3 = 0
3. 3x^2 + x + 1 = 0
4. 2x(x - 5) = -12
is there any reason to chose one method over the other to solve the equations if so which one for each?
generally, if I don't see right off that it will factor (as will #1), I check the discriminant. If it is a perfect square (as with #2) then I know it will factor, and I just look at it a bit more. If it still doesn't seem obvious, then I know I can easily use the QF to pry out the rational roots.
Other than that, only experience will give you a feel for the kinds of quadratics that yield to which method.
When solving quadratic equations, like the ones provided, there are multiple methods you can use: factoring, completing the square, or using the quadratic formula. The choice of method depends on the specific equation and your familiarity or preference with each method.
Let's go through each equation and discuss which method might be most suitable:
1. x^2 - x = 6:
To solve this equation, you can rearrange it to form x^2 - x - 6 = 0. In this case, factoring is a good option, as the equation factors nicely into (x + 2)(x - 3) = 0. From here, you can set each factor equal to zero and solve for x.
2. -2x^2 + 5x - 3 = 0:
Factoring this equation may not be straightforward, so in this case, the quadratic formula is a useful method. The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a). Simply substitute the values of a, b, and c into this formula, and solve for x.
3. 3x^2 + x + 1 = 0:
This quadratic equation doesn't factor nicely, and the constant term is not a perfect square or a multiple of the coefficient of x^2. Hence, factoring or completing the square are not suitable methods here. The quadratic formula would be the most appropriate method to solve for x.
4. 2x(x - 5) = -12:
This equation can be solved using either factoring or the distributive property. Factoring would involve factoring out common terms, while the distributive property would involve expanding the equation. In this case, using the distributive property may lead to a more direct solution: 2x(x - 5) = 2x^2 - 10x. Then, rearrange the equation to 2x^2 - 10x = -12, which can be solved using any of the methods mentioned earlier.
Overall, the choice of method depends on the equation, your comfort level with each technique, and the specific goals of the problem at hand. It's good to be familiar with each method so you can choose the most appropriate one in different situations.