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Determine the number of solutions to the quadratic

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equation, x squared plus 14x plus 49 is equal to 0.

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There's a bunch of ways we could do it.

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We could factor it and just figure out the values of x

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that satisfy it and just count them.

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That will be the number of solutions.

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We could just apply the quadratic formula.

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But what I want to do here is actually explore the quadratic

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formula, and think about how we can determine the number of

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solutions without even maybe necessarily finding them

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explicitly.

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So the quadratic formula tells us that if we have an equation

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of the form ax squared plus bx plus c is equal to 0, that the

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solutions are going to be-- or the solution if it exists is

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going to be-- negative b plus or minus the square root of b

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squared minus 4ac.

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All of that over 2a.

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Now the reason why this can be 2 solutions is that we have a

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plus or minus here.

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If this b squared minus 4ac is a positive number-- so let's

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think about this a little bit.

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If b squared minus 4ac is greater than 0,

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what's going to happen?

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Well, then it's a positive number.

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It's going to have a square root.

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And then when you add it to negative b you're going to get

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one value for the numerator, and when you subtract it from

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negative b you are going to get another

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value in the numerator.

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So this is going to lead to two solutions.

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Now what happens if b squared minus 4ac is equal to 0?

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If this expression under the radical is equal to 0, you're

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just going to have the square root of 0.

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So it's going to be negative b plus or minus 0.

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And it doesn't matter whether you add or subtract 0, you're

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going to get the same value.

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So in that situation, the actual solution of the

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equation is going to be negative b over 2a.

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There's not going to be this plus or minus, it's not going

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to be relevant.

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You're only going to have one solution.

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So if b squared minus 4ac is equal to 0, you

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only have one solution.

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And then what happens if b squared minus 4ac

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is less than 0?

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Well if b squared minus 4ac is less than 0, this is going to

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be a negative number right here and you're going to have

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to take the square root of a negative number.

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And we know, from dealing with real numbers, you can't take

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the square root.

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There is no real number squared that becomes a

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negative number.

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So in this situation there is no solutions, or no real--

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when I say real I literally mean a real

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number-- no real solution.

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So let's think about it in the context of this

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equation right here.

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And just in case you're curious if whether this

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expression right here, b squared minus 4ac,

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has a name, it does.

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It's called the discriminant.

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This is the discriminant.

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That's that part of the quadratic equation.

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It determines the number of solutions we have.

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So if we want to figure out the number of solutions for

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this equation, we don't have to go through the whole

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quadratic equation, although it's not that much work.

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We just have to evaluate b squared minus 4ac.

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So what is b squared minus 4ac?

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So b is right here, it's 14.

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So it's 14 squared minus 4 times a, which is 1, times c,

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which is 49.

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That c, right there, times 49.

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What's 14 times 14?

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Let me do it over here.

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14 times 14.

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4 times 4 is 16.

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4 times 1 is 4.

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Plus 1 is 56.

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Put a 0.

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1 times 14 is 14.

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It is 6, 9, 1.

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It's 196.

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So this right here is 196.

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And we can ignore the 1.

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What's 4 times 49?

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So 49 times 4.

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4 times 9 is 36.

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4 times 4 is 16 plus 3 is 190-- or is

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19, so you get 196.

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So this right here is 196.

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So b squared minus 4ac is 196 minus 196.

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So 196 minus 196 is equal to 0.

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So we're dealing with a situation where the

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discriminant is equal to 0.

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We only have one solution.

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And if you want, you could try to find that one solution.

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This whole part is going to be the square root of 0.

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It's just going to be 0.

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So the solution is going to be negative b over 2a.

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And negative b is-- we could just solve it.

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Negative b is negative 14 over 2 times a. a is just 1 over 2.

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So it's equal to negative 7.

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That's the only solution to this equation.

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But if you just wanted to know how many solutions, you just

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have to find out that b squared minus 4ac is 0.

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So it's only going to have one solution.

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And there's other ways.

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You could have actually factored this pretty easily

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into x plus 7 times x plus 7 and gotten the same result.

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