All Answers 357312 Answers

First, let's focus on the expression we need A and B to be equal to. 

In order to find what we need, we need to be able to compare coefficients. let's convert it to standard form. 

We have

$(x - 2)(x + 2)=x^2-4$

However, the constans don't match. Thus, we need to divide what we have by -4 to get a constant of +1. 

We have

$-\frac{1}{4}x^2+1$

Now. we cam compare coefficients. Since A is the coefficient of x^2, we have

$A=\frac{-1}{4}$

Since B is the coefficient of x, we have $B=0$

Thus, we finally have 

$A+B=-\frac{1}{4}+0 = -\frac{1}{4}$

So -1/4 is our final answer. 

Thanks! :)

Let's create a system of equations to solve this question. 

First, we know that the two points must be on the equation $x^2+y^2=25$

We also know that x must be 4. 

Thus, plugging in x=4 into the first equation, we can find y values. We have

$4^2+y^2=25\\ y^2=25-16\\ y=\pm\sqrt9\\ y=3, y=-3$

Thus, the two points A and B are $(4, 3), (4, -3)$

Since they share an x value, the difference between the y values is the distance. We have

$3-(-3)=6$

Thus, 6 is our final answer. 

Thanks! :)

We just have

$64 \cdot 3,816 = 244,224$

$244224 \pmod{ 64} \equiv 0$

244224 works. 

Thanks! :)

One ordered pair $(a,b)$ satisfies the two equations $ab^4 = 48$ and $ab^3 = 24$. What is the value of b in this ordered pair?   

                                                     ab⁴  =  48       

                                                     ab³  =  24    

divide both sides of the first by     

respective equal values of the     

second to obtain                              b = 2     

the problem doesn't ask for     

the value of a, but it's                       a = 3      

_.    

A school orders $99$ textbooks, all for the same price.   

When the bill for the total order comes, the first and last   

digits are obscured. What are the missing digits?   
$_5801.3_   

Consider that the price per book is multiplied by 99.   

And, multiplying by 99 incorporates multiplying by 9.    

The digits of the product of a number multiplied by 9    

will always sum to a total that is evenly divisible by 9.    

5 + 8 + 0 + 1 + 3 = 17  so the sum of the two missing     

numbers will have to add 1 or 10 or 19, etc. to that 17.     

Using brute force, I reckoned that 15801.39 is evenly

divided by 99.  So the missing numbers are 1 and 9.    

And the cost of each textbook was 159.61.    

_.    

First, let's move all terms to one side and combine all like terms. We get the equation

$x^{2}+10x+9=0$

Now, notice that we can easily factor the quadratic equation. We get

$(x+9)(x+1)=0$

This gives us the values for x of 

$x=-1\\ x=-9$

Thus, our answer is -1, -9.

Thanks! :)

First , simplify this equation to:

 x^2 + 15x + k = 0      Where   a = 1    b = 15    c = k 

   if the discriminant  (b^2 - 4ac)   is equal to zero, it will be a 'double root' 

             b^2 - 4ac = 0 

                15^2 - 4 (1)(k) = 0 

                  225 = 4k 

                    k = 225/4 

For the first problem:

To address the problem, we will use the properties of similar triangles since lines $DE$ and $EF$ are parallel to sides $BC$ and $CD$ of triangle $ABC$.

Since $DE$ is parallel to $BC$ and $EF$ is parallel to $CD$, triangle $ADF$ is similar to triangle $ABC$.
The sides of similar triangles are proportional, so we can set up the following proportions:

$$\frac{AD}{AB} = \frac{AF}{AC} = \frac{DF}{BC}$$

We know that $AF = 7$ and $DF = 2$. Let $BD = x$. The entire length $AB$ can be expressed as:

$$AB = AD + DF = AD + 2$$

Let’s designate $AD = x$. Hence, we have $AB = x + 2$.

Considering the structure of triangle $ADF$ in relation to triangle $ABC$, we also need to calculate $AC$ in terms of $AE$:

Since $EF$ is parallel to $CD$, triangles $AEF$ and $ACD$ are also similar, resulting in the relationship:

$$\frac{AE}{AC} = \frac{AF}{AB} = \frac{DF}{DC}$$

But for our immediate task of determining the length $BD$ just using $AF$ and $DF$, we will not utilize this second proportion at the moment.

As both triangles $ADF$ and $ABC$ are similar due to the parallel lines, we can set the ratios of the line segments as follows:

$$\frac{AD}{AB} = \frac{AF}{AC}$$

But we know $AB = AD + DF$, substituting this gives us:

$$\frac{x}{x + 2} = \frac{7}{7 + 2} = \frac{7}{9}$$

Cross-multiplying yields:

$$9x = 7(x + 2)$$

Expanding the right-hand side:

$$9x = 7x + 14$$

Subtract $7x$ from both sides:

$$2x = 14$$

Solving for $x$:

$$x = \frac{14}{2} = 7$$

Thus, we find that $AD = 7$.

Since we established $BD = DF = 2$, thus:

$$BD = 7 - 2 = 5$$

So the length of $BD$ is:

$$\boxed{5}$$

Find a six-digit multiple of $64$ that consists only of the digits $2$ and $4$.   

64 x 3,816 = 244,224     

_.   

Isn't it 100~

this is close to 1 *10^100, BUT it's actually 9.9999... * 10^99

If given     a - 1/a = 0 

                   then   a = 1/a    

Sub this into the first equation 

    a^3 - 1/a^3   =    (1/a)^3 - 1/(a^3)   =  1/a^3 - 1/a^3 = 0       

                                                               Did I miss something here??

$f(x) = \sqrt{x - \sqrt{x}}.$

This is impossible. No such 3 digit numbers exist for x.

The answer is $36$

Let's make some observations of the problem. 

First, note that triangle ABC is a 45-45-90 right triangleThis means that$AB = BC  =   12/\sqrt 2 =  6\sqrt 2$

Also, we have that

$AD = AD\\ BD = BD \\ AB = BC$

This means that triangles  ABD  and CBD  are congruent

Thus, we have $[ ABD ]   =  (1/2) [ABC]   =  (1/2) ( 1/2) ( 6\sqrt 2)^2  =  (1/4) (72)   = 18$

So our aswer is 18. 

Thanks! :)

Going form R to S : 

  X   goes from -6  to 15    a distance of 21      2/3 of this is 14       add 14 to -6 to get    x = 8

  Y   goes from 19 to -5     a distance of -24     2/3 of this is -16      add -16 to 19 to get  y = 3

(8,3) 

The coordinates of the point are (2/3*(-6) + 1/3*(15), 2/3*19 + 1/3*(-5)) = (1,11).