Decide which part of the quadratic formula tells you whether the quadratic equation can be solved by factoring.
−b b2 − 4ac 2a
Use the part of the quadratic formula that you chose above and find its value, given the following quadratic equation:
4x^2 + 6x + 2 = 0
the part of b^2-4ac is called the "determinant" since it determines the properties of your roots.
in your case b^2 - 4ac = 36-4(4)(2) = 4
so you have 2 distinct real roots.
Not only that, since 4 is a perfect square, your roots will be rational.
And since the roots are rational, I bet your quadratic WILL factor.
What a neat relationship!!
To determine whether a quadratic equation can be solved by factoring, we need to look at the discriminant, which is the part of the quadratic formula under the square root sign: b^2 - 4ac.
In this case, the discriminant is b^2 - 4ac.
For the quadratic equation 4x^2 + 6x + 2 = 0, we can see that a = 4, b = 6, and c = 2.
Now we can substitute these values into the discriminant:
Discriminant = (6)^2 - 4(4)(2)
Simplifying this expression, we have:
Discriminant = 36 - 32
So the discriminant is 4.
Now, to determine whether the quadratic equation can be solved by factoring, we need to evaluate the discriminant we just found. If the discriminant is a perfect square or zero (so the square root is a whole number or zero), then the equation can be factored. Otherwise, if the discriminant is negative, the equation cannot be factored using real numbers.
In our case, the discriminant is 4, which is a perfect square (2^2 = 4). Therefore, the quadratic equation 4x^2 + 6x + 2 = 0 can be solved by factoring.
I hope this explanation helps you understand how to determine whether a quadratic equation can be solved by factoring using the quadratic formula and evaluating the discriminant.