I’ve noticed here are around 4 ways to solve a Quadratic Equation. Factoring, Completing the square, The quadratic Formula, and graphing. With some equations, it seems that all 4 can be used for solving. But for a lot of equations, sometimes factoring doesn’t work, and you’ll have to use the Quadratic formula. Or maybe the Quadratic Formula is too slow and you need to factor instead etc. Is there an easy or quick way to tell, just by looking at a quadratic equation, which method is best for solving?
Answers
Answer:
See explanation
Step-by-step explanation:
Usually, I just find the quadratic formula the easiest to go, but here are some tips to tell if which you should use.
1. First see if you can get integer values for the roots to multiply to c and add to b.
This is the easiest way to solve a quadratic equation, and I always check for whether I can easily find the roots by multiply two numbers for c and adding them for b.
For instance, the Quadratic [tex]x^2 + 5x + 6[/tex]
I can easily see that 2 and 3 are the roots here - as [tex]2\cdot3=6[/tex] and [tex]2+3=5[/tex]. If you can do this and receive integer values, then great! I'd definitely stick with this.
2. See if you can complete the square
If you have a quadratic which is a perfect square, you'll be able to tell if c is a perfect square, and b is twice that number. This will be easiest. Any quadratic equation can be solved by completing the square, but I personally stick to this method only if I can tell it's a perfect square quadratic.
3. Stick to the Quadratic Formula
If all else fails, go with the Quadratic formula. It's not that hard and it gets the job done. I would highly discourage going with graphing, as it takes too much time and it's quite a hassle unless you have a graphing calculator.
I really hope this helped!
Answer:
In my preferred order, ...
use a graphing calculatorfactoruse the quadratic formulaStep-by-step explanation:
My personal favorite is to use a graphing calculator. If it shows the zeros are whole numbers, your solution is at hand. If it shows the zeros are recognizable decimal fractions*, the solution is also at hand. If it shows the zeros are unrecognizable as fractions, they are likely irrational. Then the vertex of the graph gets you most of the way to an exact solution.
If the quadratic's coefficients are not integers, then the quadratic formula is least likely to give difficulty.
Example
The attached is a graph of the quadratic ...
y = 2x^2 -3x -4
We can see from the x-intercepts that the roots are irrational (none are fractions with a denominator that is a factor of 2). The vertex of the graph is (3/4, -5 1/8), so the vertex form of the equation for this graph is ...
y = 2(x -3/4)^2 -(5 1/8)
Solving this for y=0, we find ...
0 = 2(x -3/4)^2 -(5 1/8)
5 1/8 = 2(x -3/4)^2 . . . . . .add the constant (vertex y-value)
(5 1/8)/2 = (x -3/4)^2 . . . . divide by 2 (the leading coefficient)
2 9/16 = (x -3/4)^2
x = 3/4 ± √(41/16) . . . . . . . add the horizontal offset (vertex x-value)
x = (3 ± √41)/4 . . . . . . . . . . simplify
If you see what we've done here, you can write down the answer from the vertex and the leading coefficient:
x = (3/4) ± √((5 1/8)/2) . . . . . vertex (3/4, 5 1/8), leading coefficient 2
__
With a little practice, factoring can be a relatively simple way to find the solution. As a preliminary step, the equation should be divided by any common factor of the coefficients, so they are all relatively prime. It helps immensely to know your times tables thoroughly. I like this method, because it can get a solution with fewer steps than using the quadratic formula.
The product of two binomials is ...
(ax +b)(cx +d) = acx^2 +(ad+bc)x +bd
The product of the leading coefficient (ac) and the constant (bd) is abcd. It has factors of ad and bc, whose sum is the coefficient of the linear term. So, if you can factor the product of the leading and trailing coefficients and find a sum of factors that is the middle coefficient, you're home free. Sometimes the factors jump out at you, and sometimes you need to list a bunch of them before you see what you need.
In our example, we would be looking for factors of 2(-4) = -8 that have a sum of -3.
-8 = (-8)(1) = -4(2) = (-2)(4) = (-1)(8)
These pairs have sums of -7, -2, 2, 7. None of the sums is -3, so factoring by integers is not possible. You can see that it is unnecessary to list the pairs with positive sums in this case.
Once you have the values ad and bc, you can see that 'a' is the greatest common divisor of ac and ad. Then ad/a = d, and ac/a = c. Now you know 3 of the 4 numbers in the binomial factors.
_____
* The Rational Root Theorem can tell you what reasonable fractions might be for your quadratic. You should be familiar with the decimal equivalents of fractions with small integer denominators.
Related Questions
Question 3
Find the domain and range of the function represented by the graph?
A. Domain: -3
Range: -4<=y<=0
B. Domain: {-3, -2, -1, 0, 1}
Range: {-3, -2, 0, 1}
C. Domain: {-4, -3, 0, 1, 2}
Range: {-4, -3, -2, -1, 0}
D. Domain: -4 <= x<=-3
Range: 0<=y<=1
Answers
Answer:
Domain: [tex]-3\leq x\leq 1[/tex]
Range: [tex]-4\leq y\leq 0[/tex]
Step-by-step explanation:
If you find the coordinate of the graph you can get the domain and range
(-3,0), (-2,-4), (-1,-2), (0,0), (1, -4)
Now that we have that find the domain and range
Domain - x coordinate
Range - y coordinate
Domain: (-3, -2, -1, 0, 1)
Range: (-4, -2, 0)
Since the domain ranges from -3 to 1 you can use the inequality [tex]-3\leq x\leq 1[/tex] to represent the domain and [tex]-4\leq y\leq 0[/tex] for the range
Nevaeh had 12 blue pencils and 18 red pencils to give out equally among the maximum number of classmates. How many red pencils will she give to each classmate?
Answers
Answer:
The answer to this problem is 3
Step-by-step explanation:
Two co-teachers, Daniel and Mathew, have to grade a set of math journals.
Daniel grades 45% of the journals and Mathew grades 55% of them.
Mathew has to grade 121 journals. How many total journals have to be graded?
Answers
Answer:
220
Step-by-step explanation:
55 percentage of x = 121
so X = 220
as simple as that
Evaluate the expression when x = 11-2
10
1.-6x4 +X +20
2. 5x2 - 9x2 + 35
3. -8x + 13x3 -10
4. -9x2 + 10x -15
5. -8x2 + 13x3 +20
6. 8x4 + 13x - 10
7.-9x + 10x4 -15
8.-18x3 + 10x2 +20
9. 7x4 + 13x2-10
10. 6x3 + 13x - 10
11. 3x2 + 13x2 -10
12. 2x4 + 13x - 10
Can someone help me please
Answers
I'm confused as to why there's a random "10" but the answer is actually pretty simple. Insert (11-210) into the problems.
Basically substitute (11-210) into the numbers with the x variable.
What is the value of x in the equation below? Negative 3 minus (negative 8) minus (negative 2) = x –13
Answers
please make me as brainliest
Answer:
A
Step-by-step explanation:
How do I do? What is the answer and why
Answers
Answer:
Step-by-step explanation:
∠ADE = 74 {Vertically opposite angles are equal}
∠AED + 127 = 180 {Straight line }
∠AED = 180 - 127
∠AED = 53
In ΔADE,
∠DAE + ∠AED + ∠ADE = 180 {angle sum property of triangle}
∠DAE + 53 + 74 = 180
∠DAE + 127 = 180
∠DAE = 180 -127
∠DAE = 53
Find the slope and y-intercept of y = 2x - 5.
I
m =
b
Answers
Step-by-step explanation:
y=2x-5 is in y=mx+b
Where m = slope
[tex]slope = m = 2[/tex]
To find y-intercept :
[tex]y = 2x - 5[/tex]
Let x =0
[tex]y = 2(0) - 5 \\ y = 0 - 5 \\ y = - 5[/tex]
A number increased by 5 is less than or equal to 26
Answers
Answer:
Equation: x + 5 ≤ 26
Solved: x ≤ 21
7)
The first floor of the Willis Tower is 21 feet high. Each additional floor is 12 feet high.
a
Write an equation for the nth floor of the tower.
b.
Find the height of the 65th floor.
Answers
Answer:
Step-by-step explanation:
If the first floor of the Willis Tower is 21 feet high. and each additional floor is 12 feet high, then the floor heights as we move from one floor to another we keep increasing by 12feets and forms an arithmetic progression as shown;
21, (21+12), (21+12+12), ...
21, 33, 45...
a) To write an equation for the nth floor of the tower, we will have to find the nth term of the sequence using the formula for finding the nth term of an arithmetic sequence.
The nth term of an arithmetic sequence is expressed as [tex]T_n = a + (n-1)d[/tex]
a is the first term = 21
d is the common difference = 33-21 = 45-33 = 12
n is the number of terms
Substituting the given parameters into the formula;
[tex]T_n = 21+(n-1)*12\\T_n = 21+12n-12\\T_n = 21-12+12n\\T_n = 9+12n[/tex]
Hence the equation for the nth floor of the tower is expressed as [tex]T_n = 9+12n[/tex]
b) To get the height of the 65th floor, we will substitute n = 65 into the formula arrived at in (a)
[tex]T_n = 9+12n\\T_{65} = 9 + 12(65)\\T_{65} = 9+780\\T_{65} = 789 ft[/tex]
Hence the height of the 65th floor is 789feets.
Write the equation for Chelsea's graph in
slope-intercept form. Make sure the slope is
written in simplest form.
Answers
Answer:
[tex]y=-4x+16[/tex].
Step-by-step explanation:
From the given graph it is clear that the y-intercept is 16.
The line passes through (-2,24) and (2,8). So, slope of line is
[tex]Slope=\dfrac{y_2-y_1}{x_2-x_1}[/tex]
[tex]Slope=\dfrac{8-24}{2-(-2)}[/tex]
[tex]Slope=\dfrac{-16}{4}[/tex]
[tex]Slope=-4[/tex]
Slope intercept form of a line is
[tex]y=mx+b[/tex]
where, m is slope and b is y-intercept.
Substitute m=-4 and b=16 in the above equation.
[tex]y=(-4)x+(16)[/tex]
[tex]y=-4x+16[/tex]
Therefore, the required equation is [tex]y=-4x+16[/tex].
-7 – 9x = -61
please help solve and show your work. thank you.
Answers
Answer:
Step-by-step explanation:
-7 - 9x = -61
-9x = -54
x = 6
Word Problem 18.) You are ordering T-shirts from a catalog. Each T-shirt costs $15. The cost of shipping is $6 no matter how many shirts you order. The total cost is $111. How many T-shirts did you order? please i need help!!!
Answers
Answer:
7 t-shirts
Step-by-step explanation:
Because you know the cost of shipping ($6) is constant no matter the quantity you order (fixed cost), you can subtract it from the total cost of the order to find the total variable cost. Next you divide the left over cost by the cost per t-shirt to get the quantity ordered.
$111 - 6 = $105
$105 / $15 per shirt = 7 t-shirts
Hope this helps!
-The Business Man
evaluate zx-xy-7 use x=-3, y=8, and z=-9*
Answers
Answer:
44
Step-by-step explanation:
1. substitute variables with their constants
(-9)(-3) - (-3)(8) - 7
2. multiply all like terms
(27) - (-24) - 7
3. Combine all like terms
27 + 24 - 7
51 - 7
= 44
Answer:
44
Step-by-step explanation:
Evaluate z x - x y - 7 where x = -3, y = 8 and z = -9:
z x - x y - 7 = -9 (-3) - -3×8 - 7
Hint: | Multiply -9 and -3 together.
-9 (-3) = 27:
27 - -3×8 - 7
Hint: | Multiply -1 and -3 together.
-(-3) = 3:
27 + 3×8 - 7
Hint: | Multiply 3 and 8 together.
3×8 = 24:
27 + 24 - 7
Hint: | Evaluate 27 + 24 using long addition.
| 1 |
| 2 | 7
+ | 2 | 4
| 5 | 1:
51 - 7
Hint: | Subtract 7 from 51.
| 4 | 11
| 5 | 1
- | | 7
| 4 | 4:
Answer: 44
who would win? Justice leage or the Teen titans, the Thundercats and the Teenage mutant ninja turtles
Answers
Answer:
JUSTICE LEAGUE
Step-by-step explanation:
Find the volume of the cone.
Either enter an exact answer in terms of or use 3.14 for radius is 6 height is 8
Answers
Answer:
Hey there!
Cone formula: [tex]\pi[/tex]hr^2/3
Plugging in the values, we get the volume is 301.59
Let me know if this helps :)
Answer:
96π
301.59
Step-by-step explanation:
The volume of a cone can be calculated using the following formula:
[tex]V=\frac{1}{3}bh[/tex]
b is the base and h is the height. The base formula for a circle (area):
[tex]b=\pi r^2[/tex]
Insert this value into the formula:
[tex]V=\frac{1}{3}\pi r^2h[/tex]
Simplify the formula:
[tex]V=\frac{\pi r^2h }{3}[/tex]
Now insert the known values:
[tex]V=\frac{\pi(6)^2(8) }{3}[/tex]
Simplify:
[tex]V=\frac{\pi*36*8 }{3}\\\\V=\frac{288\pi }{3}\\\\V=96\pi[/tex]
The volume is 96π units³ or 301.59 units³.
:Done
DJ tracks the song request she received on Friday night she noted that the number of request for country songs with two more than 4/3 times a number request for hip-hop songs they're 18 request for country songs if X represents the number of request for K-pop songs which equation represents the situation how many request for hip-hop songs did the DJ receive on Friday night
Answers
Answer: DJ receive 12 requests for hip-hop songs on Friday night.
Step-by-step explanation:
As per given , we have
[tex]\text{(Number of request country songs ) = }2+\dfrac{4}{3}\times (\text{Number of request for hip-hop songs})[/tex]
Let X represents the number of request for hip-hop songs which equation represents the situation .
If they're 18 request for country songs, then
[tex]18=2+\dfrac{4}{3}X\\\\\Rightarrow\ 18-2 =\dfrac{4}{3}X\\\\\Rightarrow\ 16 =\dfrac{4}{3}X\\\\\Rightarrow\ X=\dfrac{3}{4}\times16= 3\times4=12\\\\\Rightarrow\ X=12[/tex]
Hence, DJ receive 12 requests for hip-hop songs on Friday night.
I'm bad at math. a plan for a baseball diamond is drawn in a coordinate plane. the baseball diamond is in a shape of a square.with vertices at approximately (0,64),(64,0),(0,—64), and (—64,0). one unit in the coordinate plane represents 1 ft. what is the approximate area of the baseball diamond? what is the area of the trapezoid shown?
Answers
Answer:
Area of diamond=8192
Area of trapezoid =72 square unit
Step-by-step explanation:
Area of diamond:
find the distance between the points:
(0,64), (64,0), (0,-64),(-64,0)
d=√(x2-x1)²+(y2-y1)²
d=√64²+64²
d=64√2=90.51( rounded to the nearest hundredth)
since it is shape of square then all sides are equal
A=length*width
A=64√2*64√2
A=8192 square unit4- Area of trapezoid= ((base a +base b)/2) h
base a=3, base b=6 height=8 ( from graph count the tiles)
Area of trapezoid = ((3*6)/2 )* 8
Area=18/2*8
Area=9*8=72 square unit
The area of a diamond is 8192 square units and the area of a trapezoid is 72 square units.
What is an area?The space occupied by any two-dimensional figure in a plane is called the area. The area of the outer surface of any body is called the surface area.
The area of the diamond will be calculated as below:-
find the distance between the points:
(0,64), (64,0), (0,-64),(-64,0)
d=√(x2-x1)²+(y2-y1)²
d=√64²+64²
d=64√2=90.51( rounded to the nearest hundredth)
Since it is the shape of a square then all sides are equal
A=length x width
A=64√2 x 64√2
A=8192 square unit
The area of the trapezoid will be calculated as below:-
Area of trapezoid= ((base a +base b)/2) h
base a=3, base b=6 height=8 ( from graph count the tiles)
Area of trapezoid = ((3*6)/2 )* 8
Area=( 18/2 ) x 8
Area=9 x 8=72 square unit
Therefore, the area of a diamond is 8192 square units and the area of a trapezoid is 72 square units.
To know more about the area of the trapezoid follow
https://brainly.com/question/1410008
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Which expressions are equivalent to 4(4a+5
Answers
Answer:
4(4a+5) = 16 a + 20 = True
Step-by-step explanation:
Find the value of b×6 when b=7
Answers
Answer:
42
Step-by-step explanation:
It is basically 7×6 when you replace the b with 7
Answer:
The answer is 13.
Step-by-step explanation:
Find the value of 6 + b:
Replacing b by 7 gives 6 + 7. (example)
The formula for the area of a triangle is A=1/2 bh, where b is the base of the triangle. what is the length of the base if the area is 32cm squared and the height is 4 cm
Answers
Answer:
base =16Step-by-step explanation:
[tex]A = \frac{1}{2} \times b \times h\\A = 32\\b = ?=x\\h = 4\\\\32 = \frac{1}{2} \times x \times 4\\\\32 = \frac{4x}{2} \\\\Cross \: Multiply\\\\64 =4x \\Divde \: both \: sides \: of \: the \:equation \:by \: 4\\\\\frac{4x}{4} = \frac{64}{4} \\\\x = 16[/tex]
Which of the following is a factor of x^2 -5x-6
Answers
Answer:
(x-6)(x+1)
Step-by-step explanation:
Show your work t
please helppp!!!!!!!!
Answers
What would be the total if 25%is 18
Answers
Answer:
72
Step-by-step explanation:
You would first change 25% to 0.25
Then divide 18 by 0.25 (18/0.25)
Which equals 72
Answer:
4.5 because its basically 18 divided by 4
Step-by-step explanation:
−18+p=18 what property would be used for this
Answers
Answer:
addition property of equality
Step-by-step explanation:
−18+p=18
Add 18 to each side using the addition property of equality
−18+18+p=18+18
p = 36
is 6 greater than 0.06 and 0.006
Answers
Answer:
Yes, 6 is greater then 0.06 and 0.006
Step-by-step explanation:
6 = 6.00
and both 0.06 and 0.006 are not even a whole number
Answer: Yes it is
Step-by-step explanation:
6>0.06 and 6 >0.006
D is the midpoint of AB. If AD=2x+11 and DB=4x-5, find the value of x and AB
Answers
Answer:
x = 8
AB = 54
Step-by-step explanation:
AD = DB
2x+11 = 4x-5
11 = 2x - 5
16 = 2x
8 = x
2(8) + 11 =
16 + 11 = 27
Which expression could be used to calculate the volume of the new can?
Answers
Answer:
C.
Step-by-step explanation:
pi(r)^2
pi(2x+1)^2
6x(4pix^2+4pix+1pi)
24pix^3+24pix^2+6pix
Answer:
C
Explanation:
Volume for a cylinder is V= (pi)(r^2)(h)
r being radius, h being height.
The radius is 2x+1, so square that
(2x+1)^2 = 4x^2 + 4x + 1
Then multiply by 6x, the height
(4x^2 + 4x + 1)(6x) = 24x^3 + 24x^2 + 6x
Then multiply each term pi, which would be answer C.
Evaluate the expression using x = –1 and y=3. (6x4y3)
Answers
Answer:
6(-3)*4(3)*3
-12 *12 *3
-144*3
-432
Answer: -432
Step-by-step explanation:
Given the function f(x) = -x^2 +7x + 6, find the interval(s) in which f(x) is decreasing.
Answers
Answer:
Step-by-step explanation:
f'(x)=-2x+7
f(x) is decreasing if '(x)<0
so -2x+7<0
or 7<2x
or 2x>7
or x>7/2
including end points f(x) is decreasing in [3/2,∞)
How many solutions does this equation have? -3 – 7t = -7t - 3
1)no solution
2)one solution
3)infinitely many solutions
Answers
Answer:
3)infinitely many solution
Step-by-step explanation:
-3 - 7t = -7t - 3
-7t + 7t = -3 + 3
0 = 0
if there is same number on both side it have infinitely many solution.
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