The quadratic formula. Our goal is to use the quadratic formula to solve quadratic
equations and also to verify those solutions graphically when possible. The quadratic
formula states that the solutions of any quadratic equation in the form of a times
x squared plus b times x plus c equals zero, where a does not equal zero, are given by
x equals negative b plus or minus the square root of b squared minus four AC all over
two times a. The formula is not magic; it has been derived by completing the square on the equation
a times x squared plus b times x plus c equals zero, and this is shown in a different video.
Graphically, the zeros of a quadratic function occur where y or f of x is equal to zero. If
the zeros of a quadratic function are real, they will be the x intercepts of the parabola.
Recall that to find x intercepts, you set y equal to zero. So there is a connection
between the x intercepts and the solutions to quadratic equations, and here it is:
If you are given f of x equals a times x squared plus b times x plus c, and you set f of x equal
to zero, you would have basically a quadratic equation set equal to zero. So the solutions
to this quadratic equation will also be the x intercepts as long as they are real values.
Let’s take a look at an example. We want to solve x squared plus three times x plus ten equals
zero and check the solution graphically. This is factorable, but we are going to solve this using
the quadratic formula. The first thing we need to do is identify the values of a, b, and c. A is
equal to the coefficient of x squared, b is equal to the coefficient of x, and c is equal to the
constant ten. We are going to do the substitution into the formula and then begin to simplify.
B squared would be three squared. I'm not doing any math this time; I'm just doing the
substitution. Now let's begin to simplify. Let’s come back to the square root. Okay,
the number underneath the square root is called the discriminant. Let’s see if we can figure
out what that’s going to be. Three squared, of course, would be nine. We would have nine minus
four times one times ten, so that would become nine plus forty, or forty-nine. And it happens
that forty-nine is a perfect square, so this becomes negative three plus or minus
seven divided by negative two. So we have two solutions. The first solution x equals negative
three plus seven divided by negative two, that would give us a value of negative two. And our
second solution x equals negative ten divided by negative two, which is equal to five.
So our solutions to this quadratic equation are x equals negative two and
x equals five. Now graphically, if we let f of x equal this quadratic,
negative two and positive five should be our x intercepts. Let’s verify that. To save some time,
I’ve already typed in the quadratic equation into y one. Take a look at the graph, and this
does verify our work. We have an x intercept here of negative two and an x intercept of positive
five. I could also check the table to make sure that when x is five, I have a y value of zero,
and when x is negative two, I have a y value of zero. Our work has been done correctly.
Let’s try one more: solve two times x squared minus four times x minus three
equals zero and check your answer graphically on the graphing calculator. Okay, we’re going
to go ahead and identify our values for a, b, and c. A equals two, b equals negative four, c
equals negative three. Again, the first step I’m just going to do the substitution.
All over two times a. And again, start simplifying. Okay, so negative four becomes
positive four plus. Let’s come back to the discriminant. Our denominator is also four. Okay,
on this one, we’re going to have negative four squared, which is positive sixteen,
minus four times two times negative three, which becomes plus twenty-four. So sixteen
plus twenty-four would give us forty. We need to simplify this square root. Now remember forty;
I could write forty as four times ten, and the square root of four would be two.
So this would become four plus two times the square root of ten divided by four.
Now it’s very tempting to simplify these two fours, but that would be incorrect. I always
like to take the extra step and write this as two separate fractions to simplify it.
Four divided by four plus or minus two times the square root of ten divided by four and simplify
them individually. This will eliminate any careless simplifying mistakes. Four
divided by four will give us one, plus or minus two times the square root of ten divided by four.
Of course, simplifying two divided by four gives us one-half. So our solutions are x
equals one plus the square root of ten divided by two and x equals one minus the square root of ten
divided by two. Now it would be very difficult to try to verify these values as x intercepts
because they’re irrational, so I’ve already taken the time to convert them into a decimal,
and this would be approximately two point five eight, and this would be approximately
zero point five eight. These are the values we will look to see if these are our x intercepts.
Let’s go back to the graphing calculator. Okay, I already have the function typed into y two, so
what I’m going to do is go over to the equal sign on the first function that we just used,
hit enter to turn it off, and hit enter on y two to turn it back on. And I’m
going to graph it. It looks like this is somewhere between negative one and zero,
maybe zero point five eight, and this one is between two and three. Again,
a good sign. Let’s just go ahead and verify one of them, more specifically, if I hit second
trace, it brings up the calculation menu, and I can find the zero. I can find one of the zeros;
it’s going to ask me to go to the left side of one of the zeros. Let’s verify the one on the right.
I’m on the left side; I’m going to hit enter. I’m going to move to the right side of that zero,
hit enter, and then enter again. And it looks like that is correct;
the zero is two point five eight approximately,
which is one of the values that we found. I’ll leave the other verification to you. I hope it
helps explain how to use the quadratic formula and how to verify it graphically when possible.