In this lesson, we're going to talk about solving quadratic equations by factoring. So let's start with this example, x squared minus 49 is equal to 0. You can use the difference of perfect squares technique for this one. The square root of x squared is x, the square root of 49 is 7. So it's going to be x plus 7 and x minus 7. Now, you need to set each factor equal to 0 at this point, and then you can find the value of x.
So, we have x plus 7 is equal to 0, and x minus 7 is equal to 0. The reason why we can do that is because if one of these terms is equal to 0, then everything is 0. 0 times anything is 0. So, x is equal to negative 7. And in the other equation, if we add 7 to both sides, we can see that x is equal to positive 7. Let's try another example. Let's say if we have 3x squared minus 75 is equal to 0. What is the value of x? 3 and 75 are not perfect squares. So, we don't want to use the difference of perfect squares technique yet.
However, we can take out the GCF, the greatest common factor, which is 3. 3x squared divided by 3 is x squared. Negative 75 divided by 3 is x squared. is negative 25. Now we can use the difference of perfect squares technique to factor x squared minus 25. The square root of x squared is x, the square root of 25 is 5, so it's going to be x plus 5 and x minus 5. So, if we set x plus 5 equal to 0, we can clearly see that x will be equal to negative 5. And if we set x minus 5 equal to 0, x is equal to plus 5. And so that's it for that one.
Now what about this one? Let's say if we have 9x squared minus 64 is equal to 0. Well first... We can use the difference of perfect squares technique. We can square root 9, and we can square root 64. The square root of 9 is 3. The square root of x squared is x. The square root of 64 is 8. So it's going to be 3x plus 8, 3x minus 8. So if we set 3x plus 8 equal to 0, Then we can see that 3x is equal to negative 8, which means x is equal to negative 8 over 3. Now if we set 3x minus 8 equal to 0 and solve for x, x is going to be positive 8 over 3, using the same steps.
Now what if we have a trinomial? x squared minus 2x minus 15. And the lead-in coefficient is 1. How can we factor this expression? All you need to do is find two numbers that multiply to negative 15, but that add to negative 2. Numbers that multiply to 15 are 5 and 3. So we have positive 5 and negative 3, or negative 5 and 3. 5 plus negative 3 adds up to positive 2, but negative 5 plus 3 adds up to negative 2. So this is what we want to use. It turns out that to factor it, it's simply going to be x minus 5 plus x plus 3. So if we set x minus 5 equal to 0, x will be equal to 5. And if we set x plus 3 equal to 0, x will be equal to negative 3. Let's try another one like that. Let's say if we have x squared plus...
3x minus 28. So what two numbers multiply to negative 28 but add to 3? Go ahead and try. So if we divide 28 by 1, we'll get negative 28. 28 if we divide negative 28 by 2 negative 14 3 doesn't go into it if we divide it by 4 we'll get negative 7 4 and negative 7 differs by 3 if we add them it's negative 3 so we need to change the sign so it's going to be X minus 4 times X plus 7 which means that X is equal to positive 4 and negative 7 Here's another problem.
So how can we factor this trinomial when the leading coefficient is not 1? So what we need to do in this problem, we need to multiply 8 and negative 15. 8 times negative 15 is negative 120. Now what two numbers multiply to negative 120 but add to 2? If you're not sure, make a list.
Let's start with 1. We have 1 in 120, 2 in 120, and 2 in 120. and 60 3 and 44 and 35 and 24 6 and 28 and 15 now 10 and 10 and 12 seem promising. 10 and negative 12 differ by negative 2, but positive 12 and negative 10 adds up to positive 2. So what we're going to do in this problem is we're going to replace 2x with 12x and negative 10x. And then factor by grouping. In the first two terms, let's take out the GCF, which is going to be 4x.
8x squared divided by 4x is 2x. And 12x divided by 4x is 3. And the last two terms take out the greatest common factor, in this case, negative 5. Negative 10x divided by negative 5 is 2x. Negative 15 divided by negative 5, that's plus 3. Now, if you get two common terms, that means you're on the right track. You can write it once in a parenthesis, in the next line.
Now, the stuff on the outside, 4x, negative 5, that's going to go in the second parenthesis. So, that's what we have. Now let's set 2x plus 3 equal to 0 and 4x minus 5 equal to 0. So in the first equation, let's subtract 3 from both sides. So 2x is equal to negative 3. And then let's divide by 2. So the first answer, x, is equal to negative 3 over 2. Now let's find the other answer.
So let's add 5 to both sides. So we can see that 4x is equal to 5. And then let's divide both sides by 4. So x is equal to 5 over 4. And that's it for this problem. Now let's get some of the answers to the quadratic equations that we had in the last lesson. So for this particular problem, when we factor it, we got a solution of 5 and negative 3, and less than 10.2. But now let's use the quadratic equation to get those same answers.
So x is equal to negative b plus or minus the square root of b squared minus 4ac divided by 2. That's the quadratic formula. And you need the quadratic equation in standard form. So we can see that a is equal to 1, b is the number in front of x, b is negative 2, and c is negative 15. So let's replace b with negative 2. b squared, or negative 2 squared, negative 2 times negative 2 is 4, a is 1, and c is negative 15, divided by 2a, or 2 times 1, which is 2. Negative times negative 2 is positive 2. And then we have 4. Negative 4 times negative 15, that's positive 60. And 60 plus 4 is 64. Now the square root of 64 is 8. So we have 2 plus or minus 8. divided by 2. 2 plus 8 is 10. 10 divided by 2 is 5. That gives us the first answer.
The next one is 2 minus 8 divided by 2. 2 minus 8 is negative 6. negative 6 divided by 2 is negative 3, which gives us the second answer. So you can solve a quadratic equation by factoring or by using the quadratic formula. Now let's try another example. 8x squared plus 2x minus 15. Use the quadratic equation to find the values of x. So we can see that a is equal to 8, b is the number in front of x, that's 2, c is negative 15. So using the quadratic formula, x equals negative b plus or minus the square root of b squared minus 4ac divided by 2a.
So b is 2, which means b squared, that's going to be positive 4, minus 4 times a, a is 8, c is negative 15, divided by 2a, or 2 times 8, which is 16. So this is negative 2 plus or minus square root 4. Now negative 4 times negative 15 is positive 60. 60 times 8, that's 480. So we have 4 plus 480. So this is negative 2 plus or minus the square root of 484. The square root of 484 is 22. So now we have negative 2 plus or minus 22 over 16. So now what we're going to do at this point is separate that into two fractions. But let's just make some space first. So this is negative 2 plus 22 over 16, or negative 2 minus 22 over 16. Negative 2 plus 22, that's positive 20. And 20 over 16. Both numbers are divisible by 4. 20 divided by 4 is 5. 16 divided by 4 is 4. So the first answer is 5 divided by 4. Negative 2 minus 22. 22 is negative 24. 24 and 16 are both divisible by 8. Negative 24 divided by 8 is negative 3. 16 divided by 8 is 2. And so that's the other answer.
Negative 3 over 2. So now you know how to use the quadratic formula to solve quadratic equations.