What are the roots of the following quadratic equation: y = 7x^2 + 19x – 6?

Answer 1

Finding the roots of y = 7x^2 + 19x - 6 means finding the x values that make y = 0. In other words, we have to solve 7x^2 + 19x - 6 = 0.

One way to solve this is by factoring: We try to rewrite 7x^2 + 19x - 6 as a product like this: (__x + __)(__x + __).

The first and third blanks slots must multiply together to give the 7 from the 7x^2 term. The only way to do that (at least using whole numbers) is with a 7 and a 1. So now we have (7x + __)(x + __).

The remaining two slots must multiply together to give the -6 term. There are a few ways to do that --> 1 and -6, -1 and 6, 2 and -3, -2 and 3. And each of those pairs of numbers can be inserted into the two slots in two ways. So we simply try each one. We multiply each one out (i.e. the FOIL method). We'll definitely get the 7x^2 term and the -6 term. But only one combination will also give the 19x term in the middle. That one is the answer.

After trial and error, we find that the one that works is -2 and 3: (7x - 2)(x + 3)

If you multiply it out, you will get 7x^2 + 19x - 6.

So now we have turned the equation 7x^2 + 19x - 6 = 0 into (7x - 2)(x + 3) = 0.

This equation is saying that two numbers (namely 7x - 2 and x + 3) have a product of zero. But the only way two numbers can multiply and give zero is if one of them was zero to begin with.

Thus, either 7x - 2 = 0, or else x + 3 = 0. These linear equations are easy to solve: The first gives x = 2/7 and the second gives x = -3.

These are the two roots we were looking for.

A second way to approach this problem is to use the quadratic formula. This is a straightforward formula that gives the roots of any quadratic: x = (-b +/- sqrt(b^2 - 4ac)) / (2a)

The a, b, and c in this formula are the three coefficients of the quadratic: a = 7, b = 19, and c = -6.

So the formula gives x = (-19 +/- sqrt(19^2 - 4(7)(-6))) / (2(7)).

The part inside the square root is 529, whose square root is exactly 23.

So we have x = (-19 +/- 23) / 14. The two roots come from using the "+" for one and then the "-" for the other. x = (-19 + 23) / 14 = 4/14 = 2/7 and x = (-19 - 23) / 14 = -42/14 = -3.

No matter which way this problem is done, it is important to check that the two answers you got work. Plug each one in and verify that you get zero.

7(2/7)^2 + 19(2/7) - 6 = 0

7(-3)^2 + 19(-3) - 6 = 0

Answer 2

To find the roots of 7x² + 19x - 6 means to solve the equation

7x² + 19x - 6 = 0

and this is often done either by factoring or the quadratic formula. It is tempting to immediately jump to the quadratic formula. (Nothing wrong with that.) But a little bit of brainstorming will show that factoring is faster:

7x² + 19x - 6 = (7x + ___)(x + ____)

To figure out what goes into the blanks, ask yourself what two numbers multiply to -6? Clearly, the possibilities are {1, -6), {-1, 6}, {2, -3}, and {-2, 3}. Next we need for one of these pairs {a,b} to satisfy a + 7b = 19. Notice (-2) + 7(3) = 19. So, the pair we use is {-2, 3}:

7x² + 19x - 6 = (7x - 2)(x + 3)

Hence,

(7x - 2)(x + 3) = 0 <----> 7x - 2 = 0 or x + 3 = 0

Therefore, our solution is x = ²/₇ or x = -3.