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Solving Quadratic Equations

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Today, we're going to talk about solving quadratic equations. So we're going to look at what it means to be the solution to a quadratic equation, and then we'll do some examples.

So let's talk about what it means to be the solution of a quadratic equation. Remember the solution to a quadratic equation is sometimes called the zeroes or the roots of that equation. And when we're looking at a graph, the solution is going to be where y is equal to 0, which is the same as our x-intercepts. And so when y is equal to 0, if we're thinking about our quadratic algebraically, we have an equation with 0 for y. So we're really looking at an equation that looks like this. And so solving a quadratic equation algebraically means solving it when the entire equation is set to be equal to zero.

And we can have one or two real solutions for a quadratic equation depending on the number of x-intercepts. So this quadratic equation would have two real solutions. If it only intercepted the x-axis one time, then you would have one solution. And there is a possibility that your graph of the equation would not intersect the x-axis at all, in which case the equation would have no solution or no real solution.

So for my first example, I've got the quadratic equation x plus 1 times x minus 4 equals 0. And this is in factored form, so I can solve it using something called the zero factor property of multiplication which says that if any factor is going to be equal to 0, then the entire expression is equal to zero. So I know that either x plus 1 or x minus 4 is equal to zero, so I can write the equation x plus 1 equals 0 and x minus 4 equals zero.

So now to find my two solutions for x, I can solve these two equations. So here to solve, I'll subtract 1 from both sides, leaving you with x equals negative 1. And here I'll add 4 to both sides, leaving you with x is equal to positive 4. So x equals negative 1 and x equals positive 4 are the solutions to this quadratic equation.

So for my second example, I've got the quadratic equation x squared plus 5x equals 0. I notice that both of these terms have a common factor of x, so I can solve this quadratic equation by factoring. So I'm going to factor out an x from both terms. So here I'll have x because x times x will give me x squared. And here I'll have positive 5 because x times positive 5 would give me positive 5x. Then I'll bring down my equals 0.

Now again, I can use the zero factor property of multiplication which says that if any factor is equal to zero, then the entire expression is going to be equal to zero. So now I can write x equals 0 and x plus 5 equals 0. These are my two factors from my expression. So here I already see one of my solutions for my quadratic equation, x equals 0. And to find the second solution, I can solve this equation by subtracting 5 from both sides. Here this will cancel, and I'll be left with x is equal to negative 5 for my second solution. So I found that x equals 0 and x equals negative 5 are the solutions to this quadratic equation.

So for my last example, I'm going to solve the quadratic equation x squared minus 9 is equal to 0. Now this quadratic equation does not have an x term, so the easiest way to solve it is just by isolating the x variable using inverse operations. So I'm going to start by canceling out the minus 9, and I'll add nine on both sides to do that. Here this will cancel, and I'll have x squared is equal to positive 9.

Now I need to cancel out this 2 exponent, and I'll do that using the inverse operation of the square root. And I'll do that on both sides. When I take the square root of both sides, here, this cancels, and I'm left with x. And I know that x is going to either be equal to positive 3 or negative 3 because both positive 3 squared equals 9 and negative 3 squared equals 9. So x equals 3 or x equals negative 3 is my solution to this quadratic equation.

So let's go our key points from today. To solve a quadratic equation algebraically, let y equal 0 and solve the equation for x. The solutions to a quadratic equation are also called the roots or zeros. On a graph, the solutions are the x-intercepts of the parabola. And the zero factor property says that if any factor is equal to zero, then the entire expression is equal to zero. This property is used when solving a quadratic equation by factoring.

So I hope that these key points and examples helped you understand a little bit more about solving quadratic equations. Keep using your notes and keep on practicing and soon you'll be a pro. Thanks for watching.