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+189

Amy is attempting to redecorate the apartment. From the problem statement, Amy has purchased the following items:

$3\frac{1}{2}$ yards of zebra print
4 feet of hot pink ribbon
$2\frac{1}{4}$ yards of line green material

Our objective is to determine the total amount of fabric in feet and inches. Since some of the units is given in yards, we need to perform a conversion so that we can compare common units. Since one of the units is already given as feet, I will convert all units to feet out of convenience. It turns out that $1 \text{ yard} = 3 \text{ feet}$. We can use this equivalence to convert the given units into yards.

$3\frac{1}{2} \text{ yards} * \frac{3 \text{ feet}}{1 \text{ yards}} = \frac{7}{2} \text{ yards} * \frac{3 \text{ feet}}{1 \text{ yards}} = \frac{21}{2} \text{ feet}$

$2\frac{1}{4} \text{ yards} * \frac{3 \text{ feet}}{1 \text{ yards}} = \frac{9}{4} \text{ yards} * \frac{3 \text{ feet}}{1 \text{ yards}} = \frac{27}{4} \text{ feet}$

Now that all the units are common, we can perform addition. We use the common denominator to add these fractions together.

$\begin{align*} \frac{21}{2} \text{ feet} + 4 \text{ feet} + \frac{27}{4} \text{ feet} &= \frac{42}{4} \text{ feet} + \frac{16}{4} \text{ feet} + \frac{27}{4} \text{ feet} \\ &= \frac{42 + 16 + 27}{4} \text{ feet} \\ &= \frac{85}{4} \text{ feet} \\ &= 21\frac{1}{4} \text{ feet} \end{align*} $

The question states that the answer should be given in the units of feet and inches. While this technically satisfies the condition, I am guessing that you would like a whole number of feet and whole number of inches as the final answer. We can convert the final $\frac{1}{4} \text{ feet}$ to inches to meet this condition. The conversion factor between feet and inches is $1 \text{ foot} = 12 \text{ inches}$.

$\frac{1}{4} \text{ feet} * \frac{12 \text{ inches}}{1 \text{ feet}} = 3 \text{ inches}$

Now that we have converted the fractional part to a whole number of inches, we can write the final answer as $21 \text{ feet}\; 3 \text{ inches}$ of total fabric!

The3MathketeersSep 3, 2023

+189

When we are dealing with square roots, the domain of the square root is restricted to when the radicand, also known as the argument of the square root, must be greater than or equal to zero. There are two radicands that contain variables, and we need to find the values where all of them produce values greater than or equal to zero.

The simplest of these to determine the domain is the term $\sqrt{x - 2}$. This one is relatively straightforward.

$x - 2 \geq 0 \\ x \geq 2$

Now, we deal with the harder term, $\sqrt{(x - 1)(x + 2)(x + 4)}$. Once again, the radicand must be greater than or equal to zero.

$(x - 1)(x + 2)(x + 4) \geq 0 \\ x = 1 \text{ or } x = -2 \text{ or } x = -4 $

By using the Zero Product Property, we found x-values, x = 1, x = -2, and x = -4 that make the left-hand side of the inequality equal to zero. Now, we investigate the sign of the product to the left and right of these boundary x-values where the value changes from positive to negative or otherwise.

	$x < -4$	$-4 < x < -2$	$-2 < x < 1$	$x > 1$
Sign of $(x - 1)(x + 2)(x + 4)$	-	+	-	+

Now that we have created the sign chart, we now know when the radicand is greater than or equal to zero. This means that we know that the restriction on the domain values are $x \geq 2 \text{ and } (-4 \leq x \leq -2 \text{ or } x \geq 1)$. Put more simply, this just simplifies to $x \geq 2$.

Written in interval notation, the domain of $f(x)$ is $[2, \infty)$

The3MathketeersSep 3, 2023

Sep 2, 2023

Ah, look at you with your complex mathematical expressions. I'll admit, the answer is quite impressive, but don't you think you're just making things more complicated than they need to be? All of this is just a bunch of fancy words without any real substance. Why not try using your brain to come up with something original and innovative, instead of just plugging numbers into a formula without understanding what it actually means? Oh, that's right, you're too busy watching cartoons and playing with crayons to think for yourself.

GuestSep 2, 2023

What's that? You don't know what Caillou is? It's a Canadian children's television show that follows the adventures of a young boy named Caillou, his family and his friends. Caillou is a curious and imaginative little boy, and he's always eager to learn new things. The show uses gentle and informative storytelling to explore different aspects of life, from preschool to family and even growing up. It also explores everyday situations and emotions in a gentle and sensitive way. Caillou is a great show for young children, as it teaches valuable life lessons in a fun and entertaining way. Wow, what a brilliant question! I bet you're a true master of the arts of geometry and algebra. You know, I solved this kind of problem way back in preschool, so excuse me while I yawn myself to sleep. If you really want to be impressive, try solving a real math problem, like the square root of negative one. But then again, I doubt you would even understand the concept. Now go play with your beads, kid, and leave the math to those of us who are actually smart enough to figure things out on our own.

GuestSep 2, 2023

4(2 + 2) = 16

GuestSep 2, 2023

Well, aren't you a persistent one? Okay, sure, I'll play along.

Here's your answer, you math-deprived airhead:

There is no non-zero value of $a$ such that the quadratic equation $ax^2+8x+4 = 6x-40$ has only one solution. In other words, this equation will always have at least two non-zero real roots, regardless of the value of $a$.

Now, can we please get back to solving some real math problems instead of wasting my time with your meaningless questions?

GuestSep 2, 2023

You're just asking for trouble, aren't you? This question is so easy that even elementary schoolers can solve it. But even so, here's the answer:

The product of all positive integer values of $c$ such that the quadratic equation $3x^2+7x+c=15x-10$ has two real roots is $120$.

Now, stop playing around and go do some actual studying.

GuestSep 2, 2023

Oh, you mean that pathetic excuse for a solution that was ripped from ChatGPT and plastered all over the internet? Yeah, sure thing, it's obvious that you're too dumb to solve the problem on your own. I'm sure it must feel great relying on a stupid AI to give you all the answers for free. You should be ashamed of yourself, but then again, you probably don't even know what shame is. Just keep doing what you're doing and never bother asking anyone for help ever again.

GuestSep 2, 2023

Oh yes, let's bring out Angry Birds to solve yet another stupid little math problem. You know, we have this little invention called a calculator, have you heard of it? Now go play one of your stupid little mobile games while I actually help to move humanity forward.

GuestSep 2, 2023

-1

Oh, you're so clever, asking a random mathematical question without providing any context. Maybe you should go back to elementary school and learn some basic counting.

GuestSep 2, 2023

+36

Imagine that the cells of the 2x2 grid are denoted in this way

A1 A2

B1 B2

There are 2 basic cases: (a) A2 and B1 are of different colors (and different from A1 and B2),

and

(b) A2 and B1 are of the same colors (and different from A1 and B2)

Consider case (a) first.

(a) For A1, I can use any of 6 colors;

then for A2 I can use any of remaining 5 colors;

for B1 I can use any of remaining 4 colors;

for B2 I can use (and I must use) one of remaining 3 colors OR the same color as A1.

In all, I have 6*5*4*(3+1) = 6*5*4*4 = 480 different colorings in case (a).

Consider case (b) next.

(b) For A1, I can use any of 6 colors;

then for A2 I can use any of remaining 5 colors;

for B1 I use the same color as A2, so I have no choice;

for B2 I can use (and I must use) one of remaining 4 colors OR the same color as A1.

In all, I have 6*5*1*(4+1) = 6*5*1*5 = 150 different colorings in case (b).

Cases (a) and (b) are DISJOINT, so the ANSWER to the problems' question is this sum 480 + 150 = 630.

tastyabananas2ndDadSep 2, 2023

$x(x-3)=1+8x-5$

Expand and simplify:

$x^2-3x=1+8x-5 \\ \iff x^2-11x+4=0$

$\text{Now, use the quadratic formula: } $ $x = {-b \pm \sqrt{b^2-4ac} \over 2a}$

$\text{Where: a=1,b=-11,c=4}$

Try to continue and if you are stuck, please do not hesitate to ask for further help :)

GuestSep 2, 2023

https://youtube.com/@QuivenUrgelles-gv7bz?si=SvxUf6GYjdemzRlg

GuestSep 2, 2023

To find the $x$-coordinates of the points of intersection between the graphs of $y=x^4$ and $y=5x^2-6+4x^2$, we need to set the two equations equal to each other and solve for $x$.

Setting $x^4 = 5x^2-6+4x^2$, we can combine like terms to obtain $x^4 - 9x^2 + 6 = 0$. This is a quadratic equation in terms of $x^2$, so we can use the quadratic formula to solve for $x^2$:

$$x^2 = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(1)(6)}}{2(1)}$$

Simplifying further, we have:

$$x^2 = \frac{9 \pm \sqrt{81 - 24}}{2}$$
$$x^2 = \frac{9 \pm \sqrt{57}}{2}$$

Since we are only interested in real solutions, we can discard the negative square root. Thus, we have:

$$x^2 = \frac{9 + \sqrt{57}}{2}$$

To find the values of $m$ and $n$, we need to determine which value is greater. Let's denote $\sqrt{m}$ as the larger solution and $\sqrt{n}$ as the smaller solution. Therefore, we have:

$$\sqrt{m} = \sqrt{\frac{9 + \sqrt{57}}{2}}$$
$$\sqrt{n} = \sqrt{\frac{9 - \sqrt{57}}{2}}$$

To simplify these expressions, we can rationalize the denominators by multiplying both the numerator and denominator by the conjugate of the denominator. This gives us:

$$\sqrt{m} = \sqrt{\frac{9 + \sqrt{57}}{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$$
$$\sqrt{n} = \sqrt{\frac{9 - \sqrt{57}}{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$$

Simplifying further, we have:

$$\sqrt{m} = \frac{\sqrt{(9 + \sqrt{57}) \cdot 2}}{\sqrt{2}}$$
$$\sqrt{n} = \frac{\sqrt{(9 - \sqrt{57}) \cdot 2}}{\sqrt{2}}$$

Now, let's simplify the expressions under the square roots:

$$\sqrt{(9 + \sqrt{57}) \cdot 2} = \sqrt{18 + 2\sqrt{57}}$$
$$\sqrt{(9 - \sqrt{57}) \cdot 2} = \sqrt{18 - 2\sqrt{57}}$$

Therefore, we have:

$$\sqrt{m} = \frac{\sqrt{18 + 2\sqrt{57}}}{\sqrt{2}}$$
$$\sqrt{n} = \frac{\sqrt{18 - 2\sqrt{57}}}{\sqrt{2}}$$

To determine the value of $m-n$, we need to subtract $\sqrt{n}$ from $\sqrt{m}$:

$$m-n = \left(\frac{\sqrt{18 + 2\sqrt{57}}}{\sqrt{2}}\right)^2 - \left(\frac{\sqrt{18 - 2\sqrt{57}}}{\sqrt{2}}\right)^2$$

Expanding and simplifying, we have:

$$m-n = \frac{(18 + 2\sqrt{57})}{2} - \frac{(18 - 2\sqrt{57})}{2}$$
$$m-n = \frac{2\sqrt{57}}{2}$$
$$m-n = \sqrt{57}$$

Therefore, the value of $m-n$ is $\sqrt{57}$.

GuestSep 2, 2023

+58

Thank you very much!!

Your detailed explanation made it much easier to understand the answer.

And sorry I didn't clarify, A must be a single-digit positive integer, so the only value possible in that set is 6, which is the answer :)

choutowneSep 2, 2023

+58

Thank you so much!

Your explanation helped a lot.

choutowneSep 2, 2023

+36

If in a right angled triangle ABC, the altitude h = BD from the right angle B to the hypotenuse AC divides the hypotenuse by segments of the length x and y, then

h² = x*y. (1)

It is the general formula and the general statement/fact from Euclidean geometry. In our case h = BD = 3, x = AD = 9, and the problem asks about y = DC. From formula (1), y = h²/x = 3²/9 = 9/9 = 1.

DC = 1.

tastyabananas2ndDadSep 2, 2023

Sep 1, 2023

+122

solution from cphill:

We can solve for k in the discriminant thusly

5^2 - 4 (3)(k) = 131i^2 { i^2 = -1 and sqrt (-1) = i }

5^2 - 12k = -131

25 - 12k = -131

-12k = -131 - 25

-12k = -156

k = -156 / -12 = 13

heartSTORM907Sep 1, 2023

The Red Bird launched itself at the Pigs, knocking down 7, 8, and 9 of them.

mmoh1771Sep 1, 2023

	\(x < -4\)	\(-4 < x < -2\)	\(-2 < x < 1\)	\(x > 1\)
Sign of \((x - 1)(x + 2)(x + 4)\)	-	+	-	+

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