In this new unit we're going to be looking at Quadratic Equations and Inequalities. These are going to be equations and inequalities that have a variable squared. Now I know we already learned how to factor quadratics from before and we even used that to help us solve quadratic equations. We're going to learn some other methods that we can use to also help us solve these. Let's review factoring quadratics before we get started. How do we even know if a quadratic is factorable? If this is factorable, there must be factors of blank that sum to blank. I want you to fill in these blanks with one of the choices below. What do we know must be true, if we can factor this expression?
We know if this factorable, there must be factors of a times c which sum to our middle coefficient b. We know this because if we can factor this, it would be in this form. This first term ax squared comes from multiplying ax times cx. And our last term c comes from multiplying B times D. The middle term bx comes from multiplying Ax times D. And adding to that, B times Cx. We distribute the Ax twice, and we distribute the B twice. And really, all of these different variables represent real numbers.
Let's see if you still remember how to solve quadratic equations. See if you can factor this quadratic and then solve for the values of X. Be sure you write your answer as a set and use commas in between multiple solutions.
The solutions were negative 3 halves, and 1 6th. Nice work if you found both of these. You've really mastered factoring if you got these correct. If this gave you some trouble, you definitely want to review your factoring techniques, and try solving some quadratic equations on your own. We start factoring by finding the factor pair of negative 36 that sums to positive 16. Those two factors are negative 2 and 18. We use these as the coefficients of x to rewrite this middle term. Then we use factoring by grouping to get 2 x plus 3 times 6 x minus 1. Next, we set each of these factors equal to 0. We solve each equation for x to get x is equal to negative 3 halves, and x is equal to 1 6th. This is our solution set.
Try solving the second quadratic equation. Keep in mind you need to rearange this equation first, before you even start to solve. You want to set this equal to zero
The two solutions are negative 9 halves and positive 5. Nice algebraic thinking for getting this one correct. We start by setting this equation equal to 0 by subtracting 45 from both sides. Remember, for quadratic equations, we want to get them in the form Ax squared plus Bx plus C. Where here, these numbers are A, B and C. We want the other side of our equation to equal 0. Now we want to find factors of A times C, or negative 90, that sum to negative one. Those factors are negative 10 and postivie 9. We use these factors as the coefficients to rewrite our middle term. Then we use factoring by grouping to get these two factors. Next we set each factor equal to zero and solve for x. For the first equation, we'll get x is equal to negative 9 halves and when we set the second factor equal to 0, we'll get x is equal to positive 5. Finally we write our solution set. Nice work if you got that correct.
Alright, here's our last review problem. Try factoring this quadratic and figure out the solutions for x. Again, you'll want to write your answer as a solution set here. Have fun, and good luck.
Here, the correct answer is our negative 5 halves and positive 2. Now that's amazing if you got that correct. You've really gained steam in solving quadratic equations. We start solving this equation by subtracting 10 from both sides. This sets our quadratic equation equal to 0, and now we can try and factor. We want to find factors of negative 20 that sum to positive 1. These factors are positive 5 and negative 4. We use these coefficients to rewrite our middle term of x, and these factoring by grouping to get these factors. We set each factor equal to 0, and then solve each equation for x. This gives us a solution set of negative 5 halves and positive 2. I hope this was a great refresher on solving quadratic equations. In the upcoming lessons, we'll find other ways to solve these quadratics. Sometimes they aren't always factorable.