How do you know when to solve a quadratic equation with factoring?
Can someone please explain this to me?
If the coefficients are reasonably small, I will look for factors first.
e.g. 2x^2 - 11x - 6
You know the factors can only be (2x ....)(x ....) for the front
and for the back in can only be (.... +2)(... -3), (.... -2)(.... +3), (.... +6)(..... -1), (.... -6)(..... +1)
With a piece of scrap paper, a pencil and some common sense, or having done a few thousands of these, it can be seen to be (2x + 1)(x - 6)
A foolproof way is this:
recall that for ax^2 + bx + c = 0 , using the formula we have the expression √(b^2 - 4ac) called the discriminant.
Evaluate that. If the result is a perfect square, it WILL factor,
e.g. for our example , b^2 - 4ac = 121 - 4(2)(-6) = 169, and √169 = 13
So it will factor!
If you are on a test, and the question demands that you solve it by factoring, but you can't find the factors, do the following
1. find the answers by using the formula, remember that you are looking for the discriminant to be a perfect square.
2. In our case , x = (11 ± √169)/4 = (11 ± 13)/4 = 6 or -1/2
x = 6 ----> the factor (x-6)
x = -1/2 ----> (2x+1)
so there you have the (x-6)(2x+1)
To determine when to solve a quadratic equation using factoring, you need to consider the specific characteristics of the equation. Factoring is usually preferred when you have a quadratic equation in the standard form of "ax^2 + bx + c = 0," where "a," "b," and "c" are coefficients.
Here are the steps to determine if factoring is appropriate for solving a quadratic equation:
1. Check if the equation is in standard form: Make sure the equation is in the form "ax^2 + bx + c = 0." If it is not, rearrange the equation to bring all terms to one side.
2. Determine the value of "a": Identify the coefficient "a" in front of the x^2 term. If "a" is not equal to 1, you may need to factor out the common factor before applying factoring.
3. Check for factors of "c": Look for two numbers that multiply to give you "c," the constant term. These numbers should also add up or subtract to give you the coefficient "b" of the x term.
4. Apply factoring: If you find the factors of "c" that satisfy the condition mentioned in Step 3, you can express the quadratic equation as a product of two binomials.
5. Set each factor equal to zero: Set each binomial factor equal to zero and solve for "x" individually. This will give you two potential solutions.
6. Verify the solutions: Substitute the solutions back into the original equation to ensure they satisfy the equation. If they do, the solutions are valid.
If you cannot find factors of "c" that satisfy the conditions mentioned in Step 3, factoring may not be the most suitable method. In such cases, you can try other methods like completing the square or using the quadratic formula.