Knowing how and when to use the quadratic formula will be very helpful in Chemistry 2 or AP Chemistry when you cover concepts involving ice tables.
The quadratic formula is used to solve for x in a quadratic equation.
The generic form of a quadratic equation looks like this.
Where a, b, and c are coefficients
and the variable's powers are in descending order meaning it will always start with x squared,
then x is in the middle and this last term is just a number. And these signs can change.
To be able to solve for x in a quadratic equation all terms need to be in this descending order and set equal to zero.
Here's an example of a quadratic equation, let's review how to use the quadratic formula to solve for x.
Always start with checking if the quadratic equation is in its proper form meaning it starts with x squared, then x, then a value at the end and this is all set equal to zero.
Once this is in the right form you can identify your coefficients and label them as a,b and c,
where the first coefficient is a, the second coefficient is b and the last coefficient is c.
Make sure to include any negative signs that are in front of your coefficients.
Now that you know what a,b and c are you can plug this into the quadratic formula.
Remember when you are plugging in a number into a formula these numbers should be written in parentheses.
So plug in 4 for each b.
Next plug in 1 for each a and finally plug in negative 21 for c.
This next step goes back to order of operations where we will start with the exponents since there are no parentheses that have a math operation inside.
Four squared is 16.
Next we can multiply these two negative numbers to get a positive 84.
We can also drop this negative 4 down and multiply 2 times 1 which is 2.
Add these two numbers to get 100.
Take the square root of 100 which is 10 and now we need to separate this into two separate fractions.
We need to do this because you will actually get two x values whenever you are using the quadratic formula.
One fraction will have the plus sign while the other fraction will have the minus sign.
Solve for each x value.
Negative 4 plus 10 is 6, and 6 divided by 2 is 3. That is our first x value.
Moving on to the second fraction.
Negative 4 minus 10 is negative 14, negative 14 divided by 2 is negative 7.
These are both of our x values.
In Chemistry you will still get two x values, typically one that is positive and the other is negative.
But we only use the positive value because the x represents a concentration and we can't have a negative concentration.
You'll see this in the following equilibrium example.
This example states a 0.250 molar sample of COBr2 gas decomposes into CO and Br2.
If K is equal to 0.190, what is the concentration of each species at equilibrium?
Because this question is asking for the concentration at equilibrium
and we are only given the initial concentration of the reactant this is how you know you need to build an ICE table.
The first step of building an ICE table is writing the balanced chemical equation
which would look like this.
Next step is to write the initial concentrations for each chemical species, this represents the I in ICE.
The initial concentration for the reactant was given and it is in the correct unit which is molarity.
For ICE Tables the units must be in molarity, if it is not then you would have to convert the unit into molarity first.
As for the initial concentrations of the products those are not given so we will label them as zero for each.
The C is the change and this is something we don't know so we will label the change x,
reactants are negative since they are being used up and products are positive since they are being produced.
The E represents the reaction at equilibrium so we will add up rows I and C together to get E.
Now that the ICE table is built we can move on to writing the equilibrium constant, K.
To write an equilibrium constant we will use the balanced chemical equation and place the products on top and reactants on bottom.
When writing the equilibrium constant expression you only use chemical species that are in their aqueous or gas state.
We do not include chemical species that are solids or liquids in the equilibrium constant expression.
Also, if there were any coefficients in front then those would be the new exponents.
In this case we only have 1 coefficients in front so this is what the equilibrium constant expression would look like.
Next step is to plug in what we know,
so we can plug in the given value of K and plug in row E from our ICE table for each reactant and product.
Multiplying these x's together gives us x squared.
In order to use the quadratic formula to solve for x we first need to set everything equal to zero.
To do this we will multiply both sides by this denominator.
Distribute this value to get this.
Next we want this to be in descending order with x squared first followed by x then a number
so we must move both of these values to the opposite side by performing the opposite operations.
Now that we have the quadratic equation in the correct form,
let's identify what a, b and c are.
Plug in a, b and c into the quadratic formula.
We will square this first and multiply these values together.
Adding these two values together and taking the square root we get this.
Split this up into two separate fractions, one with the plus sign and one with a minus sign.
Starting with this fraction we will add these two values and divide by 2 to get our first x value rounded to 3 sig figs.
For our second fraction we will subtract these two values then divide by 2 to get our second x value rounded to 3 sig figs.
Since x refers to concentration the only x value that is correct is the positive x value.
Almost done, going back to our ICE table we know that the concentration at equilibrium for each chemical species would come from row E from our ICE table.
Plug in the x value we just found and solve.
So the concentration at equilibrium for our reactant would be this and our product's concentrations at equilibrium would be this.
Now what happens if after solving for x using the quadratic formula both of your x values are positive. What do you do?
Let's go over this in the next example.
We would approach this question the same way as the last, where we would start with writing the balanced chemical equation then build an ICE table.
We are given the initial concentrations of the reactants and we do not know the initial concentration of the product so we can place a zero here.
Next the change is a negative x for each reactant while our product is a positive 2x because there is a 2 coefficient in front of our product.
For row E will add rows I and C together.
Next step is to write the equilibrium constant expression.
Where the product will go on top with an exponent of 2 due to this 2 coefficient.
On the bottom we will place both reactants.
Set this equal to the given value for K and plug in row E for each reactant and product.
And now we have some math to take care of.
Start with squaring the numerator which gives us 4x squared.
Next we will foil the denominator.
Let's review how to foil.
Start with multiplying the first values together, then the outer terms, followed by the inner terms and the last terms.
Combine any like terms and this is what you get.
Going back to what we had, we need to set this all equal to zero so we will multiply both sides by this entire denominator.
These two cancel and on the left side we can distribute the 25 to each term to get this.
Combine like terms by subtracting over the 4x squared.
We want this to be in descending order so we will move each term to the opposite side which changes each one of these signs to the opposite.
Now that everything is in the proper form we can identify what a,b, and c are.
Plug this all into the quadratic formula.
Square this value and multiply this all together.
Subtract this and take the square root to get this.
Separate this into two separate fractions, one that has the plus sign and the other has a minus sign.
Starting with this fraction we will add then divide to get our first x value.
Moving on to the second fraction we will combine these values and divide to get our second x value.
Notice that both x values are positive.
This question is asking us to solve for the concentrations of each chemical species at equilibrium
If we were to plug in each x value for let's say just the first reactant we would see that one of the x's would give us a negative concentration.
This is because the x value is larger than the initial concentration and therefore is wrong.
This is how you can figure out which x value to use if both are positive.
Choose the x value that is smaller,
it has to be less than the initial concentration you were given.
This is the correct x value and these are the correct concentrations at equilibrium.
To find out when you have to use the quadratic formula or not
click the link in the description, this will lead you to the practice problems for this section that have step by step video answers.
And more helpful tips and tricks for this topic.
Try those out then come on back for the next video.