Why is the discriminant a useful tool to use when solving quadratic equations?

The roots of the quadratic equation x² - x - 56 = 0 is x = -7 or x = 8

A quadratic equation is a second-order polynomial equation in a single variable x , ax² + bx + c=0. with a ≠ 0. Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has at least one solution. The solution may be real or complex.

The roots of the quadratic equations are

x = [ -b ± √ ( b² - 4ac ) ] / ( 2a )

where ( b² - 4ac ) is the discriminant

when ( b² - 4ac ) is positive, we get two real solutions

when discriminant is zero we get just one real solution (both answers are the same)

when discriminant is negative we get a pair of complex solutions

Given data ,

Let the equation be represented as A

Now , the value of A is

x² - x - 56 = 0   be equation (1)

On simplifying the equation , we get

x² - 8x + 7x - 56 = 0

Taking the common terms in the equation , we get

x ( x - 8 ) + 7 ( x - 8 ) = 0

Now , on factorizing the equation , we get

( x + 7 ) ( x - 8 ) = 0

We have two factors which can solve the equation ,

when ( x + 7 ) = 0

Subtracting 7 on both sides of the equation , we get

x = -7

And , when ( x - 8 ) = 0

Adding 8 on both sides of the equation , we get

x = 8

Therefore , the roots of the equation are x = -7 and x = 8

Hence , the quadratic equation is solved

To learn more about quadratic equation click :

https://brainly.com/question/25652857

#SPJ1

Answer:

x = 8

x = -7

Step-by-step explanation:

To find the solutions of the quadratic equation x^2 - x - 56 = 0, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this equation, a = 1, b = -1, and c = -56. Plug these values into the formula and calculate the discriminant (the value inside the square root):

Discriminant (D) = b^2 - 4ac = (-1)^2 - 4(1)(-56) = 1 + 224 = 225

Now, calculate the two possible solutions:

x1 = (-(-1) + √225) / (2(1)) = (1 + 15) / 2 = 16 / 2 = 8

x2 = (-(-1) - √225) / (2(1)) = (1 - 15) / 2 = -14 / 2 = -7

So, the correct solutions to the equation x^2 - x - 56 = 0 are:

x = 8 (x = 0 is not a solution)

x = -7

Answer:

3

Step-by-step explanation:

x² -7x + 12 =0

by factoring:

(x-4)(x-3) = 0

x = 4

x = 3

3<4

Answer: To find the product of two matrices A and B, we must use the matrix multiplication rules.

Matrix A is a 2x2 matrix with values [2 3; 4 5] and matrix B is a 2x2 matrix with values [1 2; 3 4].

The result of matrix multiplication, AB, will also be a 2x2 matrix.

To find each element in the resulting matrix, we take the dot product of the corresponding row in matrix A and column in matrix B.

The first element in the resulting matrix AB, is found by taking the dot product of the first row in matrix A and the first column in matrix B:

(21) + (33) = 2+9 = 11

The second element in the resulting matrix AB, is found by taking the dot product of the first row in matrix A and the second column in matrix B:

(22) + (34) = 4+12 = 16

The third element in the resulting matrix AB, is found by taking the dot product of the second row in matrix A and the first column in matrix B:

(41) + (53) = 4+15 = 19

The forth element in the resulting matrix AB, is found by taking the dot product of the second row in matrix A and the second column in matrix B:

(4*2)

Step-by-step explanation:

Answer:

well, both the domain, and the range can have a positive and/or negative infinity, but as long as the 2nd factor is an opposite symbol then you will have a correct answer, for instance, -infinity, and 2 could work, but if the 2 was negative then there would be a problem

Step-by-step explanation:

Answer: 2.403 × 10^3 atoms

Step-by-step explanation:

10^13 x 6.02 = 6.02^23 and 6.02^23 x 4 = 2,403 turned into 2.403 x 10^3 atoms.

The measure of the missing angle is 17 degrees.

Trigonometry is a method that can by used to solve questions involving the application of functions. Some examples of trigonometric functions are: Sine, Cosine, Tangent, Secant, Cotangent etc.

Now considering the given question, we are to find the measure of the missing angle. We have;

cos θ = 0.9563

where θ is the missing angle the value of which is to be determined.

Thus, divide both sides of the expression by cos to have;

cos θ/ cos = 0.9563. cos

θ = cos^-1 0.9563

θ = 17.0 degrees

Therefore, the value of the missing angle is 17 degrees.

Learn more about the cosine of angles at https://brainly.com/question/3510762

#SPJ1

Answer:

To solve this problem, we can set up the following equation:

2.5 inches = 1,000 feet

We are told that the model is 18 inches high, so we can write the following equation:

18 inches = X feet

To solve for X, we can cross-multiply the equations to get:

(2.5 inches)X = (1,000 feet)(18 inches)

Simplifying this equation, we get:

X = 4,500 feet

Therefore, the height of the volcano is 4,500 feet.

Step-by-step explanation:

The unknown angle of the triangle is as follows:

∠BCE =  65 degrees

A triangle is a polygon with three sides. The sum of angles in a triangle is 180 degrees.

Therefore, the angle ∠BCE can be found as follows:

∠BAC = 70 degrees

∠ABC = 60 degrees

∠ACB = 180 - 70 - 60

∠ACB = 180 - 130

∠ACB = 50 degrees

Hence, using exterior angle theorem,

The exterior angle theorem states that the measure of an exterior angle is equal to the sum of the measures of the two remote interior angles of the triangle.

∠BCD = 60 + 70 = 130 degrees

Hence,

∠BCE = 130 / 2 = 65 degrees

learn more on angles here: https://brainly.com/question/29144629

#SPJ1

The length of ribbon is 48 yards.

The unitary technique involves first determining the value of a single unit, followed by the value of the necessary number of units.

For example, Let's say Ram spends 36 Rs. for a dozen (12) bananas.

12 bananas will set you back 36 Rs. 1 banana costs 36 x 12 = 3 Rupees.

As a result, one banana costs three rupees. Let's say we need to calculate the price of 15 bananas.

This may be done as follows: 15 bananas cost 3 rupees each; 15 units cost 45 rupees.

Given:

Kala got a prepaid debit card with $15.

Price of Ribbon = 24 cents per yard.

After purchase money left $3.48.

So, Price of Ribbon purchased = 15 - 3.48 = $11.52 = 1152 cents

length of ribbon = 1152/ 24

                           = 48 yards

Learn more about Unitary Method here:

https://brainly.com/question/22056199

#SPJ1

A) [tex]$x+y=(1+2 \sqrt{3})+(1-\sqrt{3}) i$[/tex]

B) [tex]$y+z=(\sqrt{3}+1)-(2+\sqrt{3}) i$[/tex]

c) [tex]$y-z=(\sqrt{3}-1)-\sqrt{3} i$[/tex]

are the value of given algebraic expression.

The numbers which can be expressed in the form of [tex]$x+iy[/tex] where, [tex]$x, y$[/tex] are real numbers and 'i' is an imaginary number called "iota".

here [tex]$\mathrm{i}$[/tex] is iota whose value is: [tex]$\mathrm{i}=\sqrt{-1}[/tex]

Given .

[tex]$$\begin{aligned}& x=1+\sqrt{3}+2 i \\& y=\sqrt{3}-(1+\sqrt{3}) i \\& z=1-i\end{aligned}$$[/tex]

upon solving two complex number we solve coefficient of iota together and real number together.

A) to find [tex]$(x+y)$[/tex]

[tex]$$\begin{aligned}& x+y=(1+\sqrt{3}+2 i)+(\sqrt{3}-i-\sqrt{3} i) \\& x+y=(1+2 \sqrt{3})+(2-1-\sqrt{3}) i \\& x+y=(1+2 \sqrt{3})+(1-\sqrt{3}) i\end{aligned}[/tex]

B) [tex]$\mathbf{y}+\mathbf{z}$[/tex]

[tex]$$\begin{aligned}& y+z=(\sqrt{3}-i-\sqrt{3} i)+(1-i) \\& y+z=(\sqrt{3}+1)-(2+\sqrt{3}) i\end{aligned}[/tex]

c) [tex]$\mathbf{y}-\mathbf{z}$[/tex]

[tex]$$\begin{aligned}& y-z=(\sqrt{3}-i-\sqrt{3} i)-(1-i) \\& y-z=(\sqrt{3}-1)-\sqrt{3} i\end{aligned}[/tex]

To learn more about Complex Number visit:

https://brainly.com/question/20566728

#SPJ1

The new model {A} for the total number of new green tea and fruit juice sales can be written as {A} = 9t^4 + 3t² + 105 .

What is expression ?

Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation.

Any mathematical statement that includes numbers, variables, and an arithmetic operation between them is known as an expression or algebraic expression. In the phrase 4m + 5, for instance, the terms 4m and 5 are separated from the variable m by the arithmetic sign +.

An algebraic expression is made up of a number of variables, constants, and mathematical operations. We are aware that variables have a wide range of values and no set value. They can be multiplied, divided, added, subtracted, and other mathematical operations since they are numbers. as in 3x Plus 4.

Given is that during a 20-year period, G and I can be modeled by the following equation (where t is the time in years) :

G = 6t^4+ 3t³ – 2t² + 5t + 60

I = 3t^4 - 3t³ + 5t² - 5t + 45

where {G} is the number (in billions) of new green tea sales and {I} be the number in billions of new fruit juice sales.

A new model {A} for the total number of new green tea and fruit juice sales can be written as -

{A} = {G} + {I}

{A} = 6t^4+ 3t³ – 2t² + 5t + 60 + 3t^4 - 3t³ + 5t² - 5t + 45

{A} = 9t^4 + 3t² + 105

Therefore, the new model {A} for the total number of new green tea and fruit juice sales can be written as {A} = 9t^4 + 3t² + 105

To learn more about expression from the given link  

https://brainly.com/question/15775620

#SPJ1

To calculate the distance between two points:

We apply the following formula:

d = √(x₂ - x₁)² + (y₂ - y₁)²

The points are:

We substitute data and solve:

[tex]\bf{d=\sqrt{(6-(-3))^2+(-8-8(-8))^2 } }\\ \\ \bf{ d=\sqrt{9^2+0^2}=\sqrt{81+0} }\\ \\ \bf{d=\sqrt{81}=9 }[/tex]

The distance between the points (−3,− 8), (6, – 8) is 9.

The given quadratic equation has two possible roots as -

{x} = -1/2 and {x} = 1.667.

A quadratic equation is a equation that is of the form -

y = f{x} = ax² + bx + c

Given is the statement as -

"12 x squared minus 14 x equals 10"

The given function can be written as -

f(x) = 12x² - 14x - 10

Refer to the graph attached. The x - intercepts show the solution of the given quadratic equation. So, we can write -

(x + 1/2)(x - 1.667) = 0

(x + 1/2) = 0     and     (x - 1.667) = 0

x = -1/2     and      x = 1.667

Therefore, the given quadratic equation has two possible roots as -

{x} = -1/2 and {x} = 1.667.

To solve more questions on quadratic equations, visit the link below -

brainly.com/question/22412845

#SPJ1

Therefore , as a result, x = 53 and y = 10 are the answers to the trigonometry problem that was presented.

Trigonometry is the discipline of mathematics that studies correlations between angles and sides of triangles. Trigonometry is present in all of geometry because every ahead and move form may indeed be resolved to a set of triangles. Amazingly intricate connections exist between geometry and other areas of mathematics, particularly arithmetic, real or abstract, infinite series, arithmetic, and infinite series.

Here,

Sin 30 = Opposing Side/High

1/2 = 5 / y

y = 5 / (1/2)

y = 10

Cos 30 = Adjustment Side/ Hyp

√3/2 = x/y

√3/2 = x/10

x = 10√3 / 2

x = 5√3

In light of this, the given fractional problem's answer is

x = 5√3 and y = 10.

Therefore , as a result, x = 53 and y = 10 are the answers to the trigonometry problem that was presented.

To know more about trigonometry , visit

brainly.com/question/12068045

#SPJ1

=======================================================

Explanation:

I'm assuming your teacher meant to say "same number or letter" instead of "same number of letter" at the very end of the problem.

The best case scenario is that the person selecting the cards is very lucky to select say a king of hearts and king of clubs. This shows that 2 cards is the lowest amount of cards to select here.

----------

Of course, such a scenario is not that likely. That person is likely to get some other card compared to the 1st selection.

Let's assume the person selecting the cards has the worst luck possible. This means they select each card 1,2,3,...,10,J,Q,K exactly once in any order. If this happens, then the person does not have any pairs just yet. The person has selected 13 cards so far with this worst case scenario.

The 14th card is guaranteed to pair up with some other card previously chosen. There simply aren't any other options left once you've exhausted all possibilities. See the pigeonhole principle for more information.

So this is why you would need to select 14 cards to guarantee having at least two pair up together (i.e. you would have 2 or more pairs).

Answer: I'm sorry if it's wrong I'm pretty sure it's ≥ if not I'm sorry

Step-by-step explanation:

1. 60% of 330 would fill 6 of the boxes.

2. 45% of 600 would fill 4 and half of the boxes.

3. 42% of 450 is 189.

4.  79% of 800 is 632.

A percentage is a value per hundredth. Percentages can be converted into decimals and fractions by dividing the percentage value by a hundred.

1. The strip has 10 boxes and totals to 330.

Now, 60% of 330 would fill 6 out of those 10 boxes.

2. The strip has 10 boxes and totals to 600.

So, 45% of it would fill 4 and a half of the boxes.

3. 42% of 450 can be numerically expressed as,

(42/100)×450.

= 189.

Similarly, 4. 79% of 800 is,

(79/100)×800.

= 632.

learn more about percentages here :

https://brainly.com/question/24159063

#SPJ1

on solving the equation, we have therefore the value of x from the given equation comes out to be 13.5

An equation is a formula in mathematics that joins two statements with the equal symbol = to represent equality. The definition of an equation in algebra is a mathematical statement proving the equality of two mathematical expressions. In the equation 3x + 5 = 14, for instance, the terms 3x + 5 and 14 are separated by an equal sign. The link between two phrases on either side of a letter is expressed mathematically. There is often only one variable, which is also the symbol. instance: 2x - 4 Equals 2.

equation given is 3(x+2)=7x-48

value of x

[tex]3(x+2)=7x-48\\ 3x+6=7x-48\\7x-48-3x-6=0\\ 4x-54=0\\ 4x=54\\ x=54/4\\ x=13.5\\[/tex]

value of x from the given equation comes out to be 13.5

To know more about equation visit:

https://brainly.com/question/649785

#SPJ1

Answer:

[tex](\frac{5}{3} ;\frac{4}{3} ).[/tex]

Step-by-step explanation:

[tex]\left \{ {{x+y=3} \atop {\frac{y+2}{2} =x}} \right. \ = > \ \left \{ {{x+y=3} \atop {y+2=2x}} \right. \ = > \ \left \{ {{x+y=3} \atop {2x-y=2}} \right. \ = > \ \left \{ {{x=\frac{5}{3} } \atop {y=\frac{4}{3} }} \right.[/tex]

The prices of the drum that will reach over $6500 are between $3 and $8

And the inequality expression is (50 + 4x) (120 - 5x) > 6500

From the question, we have the following parameters that can be used in our computation:

Number of models = 50

Initial price = $120

Also from the question, we have:

So, we have

Models = 50 + 4x

Price = 120 - 5x

This means that the revenue function is

R(x) = (50 + 4x) *  (120 - 5x)

When the price reaches over $6500, we have

(50 + 4x) *  (120 - 5x) > 6500

Using a graphing calculator, we have

2.911 <x < 8.5

Approximate

3 ≤ x ≤ 8

Hence, the model reaches $6500 between 3 and 8

Read more about inequality at

https://brainly.com/question/25275758

#SPJ1

Answer:

Step-by-step explanation:

Its A

Answer:

D

Step-by-step explanation:

Answer:

C.  meters

Step-by-step explanation:

The time required for Latisha to walk around the rim at a rate of 2 mph is 0.235 hours.

Speed is defined as when an object is in motion, the distance covered by that object per unit of time is called speed.

here,
Ubehebe crater in Death Valley has a diameter of a little more than 1/2 a mile.
Circumference of the rim,
= πD

= 3.14 × 1/2
= 1.57 mile

Time = Distance/speed
Time = 1.57 / 2
Time = 0.785 hours

Thus, the time required for Latisha to walk around the rim at a rate of 2 mph is 0.235 hours.

Learn more about speed here:
https://brainly.com/question/7359669

#SPJ1

Answer:

2.66

Step-by-step explanation:

Answer:

2.66

Step-by-step explanation:

look at first at tenths for the biggest no .if there is a tie proceed to hundredths for biggest no and so on

280 AND 245 RS  the company GAIN,20% OFF it is 280 rs and 30% off it is 245 rs

To learn more about discount price refers to:

brainly.com/question/7459025

#SPJ1

let the third side be b then use the pythagoras theorem

x=125

...............