How to Find the Missing Roots in a Quadratic Equation
This video shows you how to find the missing roots of the quadratic equation. The root is the solution to the quadratic equation, which is also the x intercept (where the parabola crosses the x axis.)
Finding the Missing Roots
The quadratic equation Y equals ax squared plus 2x minus 8, has -2 as one of its roots. Find the other root. The root is also the solution to the quadratic equation. -2 is one solution to this quadratic equation, that’s the x value.
It’s a solution, it’s a root, it’s also the x intercept. It’s where the parabola crosses the x axis. That means that the ordered pair would be -2,0. Y would have to be 0 in order to be on the x axis. We can use that order pair, -2,0, to substitute those values for x and y in our quadratic equation, so that we can find a.
That’s what we’re going to start, by substituting the ordered pair -2,0 into this quadratic equation. Y is 0, equals a is what we don’t know, what we’re trying to find, times x is -2 squared, plus 2, times -2, minus 8. Now when we solve this we’ll find a.
0 is equal to a times -2 squared is 4, 2 times -2 is -4, so minus 4, minus 8. 0 equals 4a, -4 minus 8 is -12. Now we solve for a, add 12 to both sides, 0 plus 12 is 12, equals 4a. -12 plus 12, those are additive inverses so they cancel.
Divide both sides by 4, 4 divided by 4 is 1, 12 divided by 4 is 3, so 3 is equal to a. Now we know what our quadratic equation is, it’s y equals 3x squared, plus 2x, minus 8. With that information we can factor out equation to find the other root.
We already know that one of our roots is x plus 2. One of our factors is x plus 2, since our route was negative 2. That means the related linear factors x plus 2. In order to foil our 2 linear factors together and get 3x squared plus 2x minus 8, this has to be 3x.
Because x times 3x when we foil will give us the 3x squared. These last numbers have to multiply to give us -8. 2 times -4 would be -8. We can foil -just quickly foil- to see if we factored correctly.
X times 3x is 3x squared, x times -4 is -4x, then 2 times 3x is 6x, so -4x plus 6x gives us the middle term, positive 2x, and then 2 times -4 gives us -8. Now we found the other linear factor, we still haven’t found the root yet though.
To find the root we must set it equal to zero and solve. 3x minus 4 equals 0. Solve for x, add 4 to both sides, additive inverses cancel, so 3x equals 0 plus 4 is 4, divide the science by 3, and so x equals 4/3. Our 2 roots are 4/3, and the other one we were given was x equals -2.
These are the two places for our parabola crosses the x axis. Those are two solutions to our quadratic equation that we were given.