Before going to solve the equations with radicals, check the index to make sure you are alike so you can add, subtract, multiply, divide what ever you need to do for the equation.
When you multiply an equation and there’s two sets of brackets, you foil the equation.
The numbers under the root sign are able to multiply together and change what the value under the root sign, when there is a coefficient that number will be multiply by the coefficient of the radical your multiplying by.
In the example the equation is solved by FOIL
The radicands can only change when multiplied with other radicands, and the coefficients can only change by multiplying by other coefficients
First terms – Multiply the first term in each of the brackets together
outside terms- Multiply the outside terms in each of the brackets together
inside terms- Multiply the inside terms in each of the brackets together
last terms- Multiply the last terms in each of the brackets together
Once you multiplying each number by each other collect like terms to simplify and make sure the number under the root sign is simplified
ex. if there was an equation where it was 5 root 9 it would turn into 5 times 9 and it would equal to 45 then you may be able to collect like terms with another number depending on the equation
Once everything is simplified and you have collected like terms and double checked you will have your answer
This week we learned how to find the line of symmetry from a trinomial
The photo shows how to find the axis of symmetry:
1) you need to simplify the trinomial and here you would divide but the coefficient of x^2 which is 3
2) After the equation is simplified which should be 3(x^2 + 2x -24) = y you should look if you can factor this and in this case you can. This factors to 3(x+6)(x-4)
3) with the factored form you are now able to find the two x values of the parabola, to do that you need to create zero in the brackets. What could x be to make the value in the brackets zero. For the first set of brackets it would be x=-6 and the second would be x=4 because when those numbers are added to the ones with them in the brackets it creates zero.
Since you have those two values which are the x values you can find the axis of symmetry.
4) to find the axis of symmetry you just add the two x values together and divide them by two. So -6+4 which is 2 than 2 divided by 2 is 1.
Analyzing Quadratics can give you the steps you need to draw a parabola on a graph.
y=a(x-p)^2+q — this is the formula of vertex/standard
With this formula you can find out…
The stretch value |a|>1 / compression |a|<1 and if it opens up or down which you can get from the “a” value. If it is a negative number the parabola would open down and if it was positive it would open up.
The “p” value tells you the horizontal translation — if the parabola is moved to the left or to the right. The “p” value is also the axis of symmetry which is the middle of the parabola
The “q” value is the vertical translation, it tells you if the parabola is moving up or down on the graph
Now the most important part of the parabola is the vertex. With this formula you can find the vertex by solving the values of “p” and “q” if you have the values for those then that will give you the vertex.
For example the question in the photo above is…
The “p” value is 5 and the “q” value is -3 in this case the vertex coordinates would be (5,-3). The vertex can be minimum or maximum and the vertex is important because it’s the highest lowest point of the parabola.
1) Find numbers that go with a, b, c
2) plug those numbers into the quadratic formula
3) Solve the steps needed in the formula, since the first step is x = -b, it is -b + 2, so it’s -2. The next step is root b^2 – 4ac, which is 2^2-4(-)(-1) divided by 2(1), solving that equals to root 4+4 which is 8, and then the whole equation is divided by 2.
4) the equation should look like x=-2 +/- root 8 all over 2
Looking at the equation you may notice root 8 can be changed into a mixed racial and will be able to simplify
convert root 8 into a mixed radical
5) the equation is now -2 +/- 2 root 2 all over 2
there is 2’s all over this equation, the number you need to divide by is 2. So simplify the equation by dividing everything by 2.
6) by doing so, the equation should now look like… x= -1 +/- root 2
that will be your final answer because x is alone and the equation on the right side of the equal sign can’t be further simplified
Completing Perfect Square
1) The constant on the left side of the equation should be moved over to the other side of the equal sign where the 0 is. You are able to do that by subtracting the 12 from the left sign of the equation and doing the same to the right. 12 now is canceled out from the left side of the equation
2) Now the equation is x^2 + 10x = -12
here you are creating a new constant in a way, you divide the 2nd term (the number attached to x not x^2) into two, then multiply those numbers together which equal to 25. Now 25 has been added to the left side… and WHATEVER YOU DO TO ONE SIDE YOU DO TO THE OTHER. So 25 needs to be added to the right side of the equal side, so that’s -12 + 25 which equals to 13.
3) Now the equation is x^2 + 10x + 25 = 13,
looking at this equation you may notice it is factorable, if you look at 10 (5+5=10) those 5’s that add into 10, are also able to go into 25 if multiplied. (5 • 5 = 25). Since this is factorable just simplify the equation by doing so.
4) the equation now is (x+5)^2 = 13
you want to get rid of the brackets and exponent, so do the opposite. Root the whole equations even the right side.
5) The equation is now x + 5 = +/- root 13
Get rid of 5 on the left by subtracting it and doing the same to the left side of the equation to get X alone
6) x = -5 +/- root 13
that is your final answer.
A Rule To Remember – Simplifying radicals
When simplifying radicals, you want to collect like terms and if there is not any then you can turn mixed radicals to entire radicals or entire radicals to mixed radicals than see for like terms.
Here there is like terms, but there is also numbers you can simplify further. Which is root 4 and root 25, when you root them they become 2 and 5. Those numbers you have to remember to multiply them by the coefficient.
Once you’ve multiplied them then collect like terms to solve the simplified equation.