TenMarks teaches you how to use the discriminant to find the number of solutions.
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Learn about the Discriminant in Quadratic Equations Using the discriminant, some of these problems we need to find the number of solutions of each equation. So before we go had and find the solutions to these equations let's quickly talk about quadratic equations. So quadratic equations it will have 2, 1 or no real solutions. We can determine the number of solutions by evaluating the discriminant. So we determine the number of solutions by evaluating the discriminant. So, if a quadratic equation and standard form, remembers standard form is ax²+bx+c=0. So if a quadratic equation is in standard form it’s discriminant than is b²-4ac. So if it’s in standard form, if a quadratic equation is in standard form its discriminate is b²-4ac. However, if b²-4ac is greater than 0 then, the equation will have two real solutions. If b²-4ac is equal to 0 then the equation has one real solution. So that means if b²-4ac is less than 0 then there are no real solutions. So let's go ahead and use this and solve our equations, so let's go to our first equation a²+bx4x+3=0. So when solving the first step is to see it’s in standard form. So I go ahead and check standard form is in this form ax²+bx+c=0 and it isn’t standard form so I don’t have to do anything. The next is I need to identify what my a is, what my b and my c. So a in this formula or in this expression is 1, b is 4 and c is 3. All right, now I need to evaluate the discriminant, so the discriminant in this would be, remember when you go down your discriminant in standard form is b²-4ac. So my discriminant b²-4ac would equal, I go ahead and plug it in, so it equal b is 4 so it’s 4²-4×a, which is 1×c which is 3. Now, I just go ahead and simplify it so a² is 16 and 4×3×1 is 12, so 12-16 is 4. So b²+4ac=4. So this is positive so there are two real solutions. There are two real solutions to this problem. Because remember if it’s greater than zero, if it’s positive there's two real solutions. Let's go ahead and move on to our second problem. So here we have 2x²+4x+3=0, our first step is to write the expression of standard form. Well, if I look it’s already in standard form so I don’t have to do anything to this so. So now I need to identify what my a is, my b and my c. So in problem a is 2, b is 4 and c is 3. Now, I'm going to go ahead and evaluate my discriminant. So again, the discriminant is b²-4ac so b²-4ac and I'm going to go ahead and plug in my integers here, right. So b is 4 so b4²-4×a, which is 2×c which is 3 and now I'm going to go ahead and simplify 4×4 is 16-4×3×2 is 24. And then that gives me -8. So, b²-4ac is negative 8, which is less than 0. So if the discriminant is less than 0 or a negative there's no real solution. So let's go ahead and move on to our third problem C. So here we have x²+4x+4=0, so again, our first step is to make sure it’s written in standard form and this is already written form so we don’t have to worry. Next, third step is to identify our a, b and c. So a is 1 and b and c, so a is 1, b again is 4 and c is 4. Now we evaluate our discriminant so again our discriminant is b²-4ac and that equals we’re going to ahead and plug everything in so b is 4²-4×a which is one times c which is 4 and we’re going to simplify and that 16 and that gives me 0. So b²-4ac is 0 and if I look, if the discriminant is equal to 0 then there is one real solution. So here we only have one real solution. So things to keep in mind, is that the number of solutions of a quadratic equation can be determine by evaluating the discriminant. So if the value to discriminant to find the number solutions, so if the quadratic equation is in standard form it’s discriminate is b²-4ac. And then, if b²-4ac is greater than 0 or positive number there are two real solutions. If the discriminant is equal 0 there is one real solution and if the discriminant is less than 0 or negative there are no real solutions.
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