Question

Identify all factors of the expression 12x^2-14x-6

Answer 1

To find all factors of the quadratic expression 12x^2-14x-6, one can use the quadratic formula since factoring by grouping is not easily applicable. The solutions from the quadratic formula will give the factors of the expression when it is set to zero.

Step-by-step explanation:

To identify all factors of the expression 12x2-14x-6, we can use methods such as factoring by grouping, the quadratic formula, or completing the square, provided that the expression is equated to zero (i.e., part of a quadratic equation). First, let's look at the expression as a quadratic equation: 12x2-14x-6 = 0.

For factoring by grouping, we would look for two numbers that multiply to give the product of the coefficient of x2 (12) and the constant term (-6), which is -72, and also add up to the coefficient of x (-14). However, since there are no two such integers, this method will not work easily in this case.

We can turn to the quadratic formula, which states that for an equation of the form ax2 + bx + c = 0, the solutions for x can be found using the formula x = [-b ± sqrt(b2 - 4ac)]/(2a). In our case, a = 12, b = -14, and c = -6. Applying the quadratic formula would give us the two solutions for x, which represent the factors of the quadratic expression when set to zero.

Lastly, one might model this expression in a way akin to a perfect square trinomial or multiply both sides by an appropriate value to reduce fractions or decimals, as per the provided reference material. However, for the given expression, these approaches would be less straightforward than using the quadratic formula.

After finding the values for x that satisfy the equation, we would express the original quadratic equation as the product of two binomial expressions that would give us the factored form of the quadratic expression.

Answer 2

Let's solve this by using the quadratic formula:

`(-b+-√(b^2-4ac) )/(2a)`

Note that we only use the coefficients so a=12, b=-14, and c=-6.

Plug values in the quadratic equation:

`\frac{ -  ( - 14)± \sqrt{ {( - 14)}^(2) - 4(12)( - 6) } }{2(12)}`

And so by evaluating those values we obtain:

`(14+-√(484) )/(24)=(14+-22)/(24)  \\\\`

Now we have two answers which are our factors one where we add another where we subtract and so:

First factor:

`(14+22)/(24)=(36)/(24)=(3)/(2)`

Second Factor:

`(14-22)/(24)=(-8)/(24)=-(1)/(3)`

And so your factors are

`(3)/(2),-(1)/(3)`

meaning that those are your roots/x-intercepts.