💡Okay, so I have this problem here where I need to find the standard deviation of five numbers: 3, a, 4, b, and 5. The catch is that a and b are the roots of the quadratic equation x² - 3x + 2 = 0. Hmm, okay, let's break this down step by step.First, I remember that to find the standard deviation, I need to follow a few steps: find the mean of the numbers, then calculate the variance by finding the average of the squared differences from the mean, and finally take the square root of the variance to get the standard deviation. But before I can do any of that, I need to figure out what a and b are because they're part of the data set.Alright, so the equation given is x² - 3x + 2 = 0. I think I can solve this quadratic equation to find the roots, which are a and b. Let me recall how to solve quadratic equations. One method is factoring, another is using the quadratic formula. Let me try factoring first because it might be quicker.Looking at the equation x² - 3x + 2, I need two numbers that multiply to 2 and add up to -3. Hmm, let's see... The factors of 2 are 1 and 2. If I take both as negative, they multiply to positive 2 and add up to -3. So, yes, that works. Therefore, the equation can be factored as (x - 1)(x - 2) = 0. Setting each factor equal to zero gives the solutions x = 1 and x = 2. So, that means a and b are 1 and 2. I think it doesn't matter which one is a or b because they're just roots, so the order doesn't affect the standard deviation calculation. So, now I can list all five numbers: 3, 1, 4, 2, 5.Next, I need to find the mean of these numbers. The mean is calculated by adding all the numbers together and then dividing by how many numbers there are. So, let's add them up: 3 + 1 + 4 + 2 + 5. Let me do that step by step to avoid mistakes. 3 + 1 is 4, plus 4 is 8, plus 2 is 10, plus 5 is 15. So, the total is 15. There are 5 numbers, so the mean is 15 divided by 5, which is 3. Okay, the mean is 3.Now, onto calculating the variance. Variance is the average of the squared differences from the mean. So, for each number, I need to subtract the mean and then square the result. Then, I'll take the average of those squared differences.Let's list the numbers again: 3, 1, 4, 2, 5. The mean is 3, so let's compute each deviation:1. For the first number, 3: 3 - 3 = 0. Squared, that's 0² = 0.2. For the second number, 1: 1 - 3 = -2. Squared, that's (-2)² = 4.3. For the third number, 4: 4 - 3 = 1. Squared, that's 1² = 1.4. For the fourth number, 2: 2 - 3 = -1. Squared, that's (-1)² = 1.5. For the fifth number, 5: 5 - 3 = 2. Squared, that's 2² = 4.Now, I have the squared differences: 0, 4, 1, 1, 4. Let's add these up to find the total squared difference. 0 + 4 is 4, plus 1 is 5, plus 1 is 6, plus 4 is 10. So, the total squared difference is 10.Since variance is the average of these squared differences, I divide by the number of data points, which is 5. So, variance = 10 / 5 = 2.Finally, standard deviation is the square root of the variance. So, standard deviation = √2. That's approximately 1.414, but since the question doesn't specify rounding, I'll leave it as √2.Wait, let me just double-check my calculations to make sure I didn't make any mistakes. The roots were 1 and 2, correct? Plugging back into the equation: (1)² - 3(1) + 2 = 1 - 3 + 2 = 0, and (2)² - 3(2) + 2 = 4 - 6 + 2 = 0. Yep, that's right.Adding the numbers: 3 + 1 + 4 + 2 + 5 = 15, correct. Mean is 15 / 5 = 3, that's solid. Squared differences: 0, 4, 1, 1, 4. Adding those gives 10, which divided by 5 is 2, so variance is 2. Square root of 2 is indeed the standard deviation.I think that's all. I don't see any mistakes in my reasoning. So, the standard deviation is √2.**Final Answer**The standard deviation of these five numbers is boxed{sqrt{2}}.

💡Alright, so I need to find the focus of the parabola given by the equation ( y = -2x^2 - 4x + 1 ). Hmm, okay, let's see. I remember that the focus of a parabola is a specific point inside the parabola that helps define its shape. But to find it, I think I need to rewrite the equation in a different form, maybe the vertex form. First, let me recall the standard form of a parabola. It's usually written as ( y = ax^2 + bx + c ), which is what I have here. In this case, ( a = -2 ), ( b = -4 ), and ( c = 1 ). Since the coefficient of ( x^2 ) is negative (( a = -2 )), the parabola opens downward. That makes sense because if ( a ) is negative, the parabola will have a maximum point at its vertex and will open downward.Now, to find the focus, I think I need to find the vertex first. I remember that the vertex form of a parabola is ( y = a(x - h)^2 + k ), where ( (h, k) ) is the vertex. So, I need to complete the square for the quadratic equation to convert it into vertex form.Starting with the given equation:[ y = -2x^2 - 4x + 1 ]I can factor out the coefficient of ( x^2 ) from the first two terms:[ y = -2(x^2 + 2x) + 1 ]Okay, now I need to complete the square inside the parentheses. The coefficient of ( x ) is 2, so I take half of that, which is 1, and square it to get 1. So, I add and subtract 1 inside the parentheses to maintain the equality:[ y = -2[(x^2 + 2x + 1 - 1)] + 1 ][ y = -2[(x + 1)^2 - 1] + 1 ]Now, distribute the -2:[ y = -2(x + 1)^2 + 2 + 1 ][ y = -2(x + 1)^2 + 3 ]Alright, so now the equation is in vertex form:[ y = -2(x + 1)^2 + 3 ]This tells me that the vertex of the parabola is at ( (-1, 3) ). Next, I need to find the focus. I recall that for a parabola in the form ( y = a(x - h)^2 + k ), the focus is located at a distance of ( frac{1}{4|a|} ) from the vertex along the axis of symmetry. Since the parabola opens downward, the focus will be below the vertex.Calculating the distance:[ frac{1}{4|a|} = frac{1}{4 times 2} = frac{1}{8} ]So, the focus is ( frac{1}{8} ) units below the vertex. The vertex is at ( (-1, 3) ), so subtracting ( frac{1}{8} ) from the y-coordinate:[ y = 3 - frac{1}{8} = frac{24}{8} - frac{1}{8} = frac{23}{8} ]Therefore, the focus of the parabola is at ( (-1, frac{23}{8}) ).Wait, let me double-check my steps to make sure I didn't make any mistakes. Starting from the original equation, I factored out -2 correctly and completed the square by adding and subtracting 1 inside the parentheses. Distributing the -2 gave me the correct vertex form, and identifying the vertex as (-1, 3) seems right. Calculating the distance for the focus, I used the formula ( frac{1}{4|a|} ), which is correct because ( a ) is -2, and taking the absolute value gives 2. So, ( frac{1}{8} ) is the correct distance. Since the parabola opens downward, moving down from the vertex by ( frac{1}{8} ) units gives the focus at ( (-1, frac{23}{8}) ). I think everything checks out. I don't see any errors in my calculations or reasoning. So, I'm confident that the focus is indeed at ( (-1, frac{23}{8}) ).

💡First, I need to determine how much money Jasmine has in total after mowing one lawn. She starts with 9.65 and earns an additional 5.35, so her total amount is 9.65 + 5.35, which equals 15.00.Next, I need to find out how many trading cards she can buy with 15.00. Each trading card costs 0.95. To find the maximum number of cards she can purchase, I divide the total amount of money she has by the cost per card: 15.00 ÷ 0.95. This calculation gives approximately 15.789.Since Jasmine cannot buy a fraction of a trading card, I round down to the nearest whole number. Therefore, she can buy 15 trading cards.Finally, I conclude that the maximum number of trading cards Jasmine can buy is 15.

💡First, calculate the cost of each type of fruit:- Grapes: 8 kg * 70/kg = 560- Mangoes: 9 kg * 55/kg = 495- Oranges: 5 kg * 40/kg = 200- Strawberries: 4 kg * 90/kg = 360- Pineapple: 6 kg * 45/kg = 270Add these amounts to find the total cost before discount and tax:Total cost = 560 + 495 + 200 + 360 + 270 = 1885Apply the 10% discount:Discount amount = 10% of 1885 = 0.10 * 1885 = 188.50Subtract the discount from the total cost:Discounted total = 1885 - 188.50 = 1696.50Calculate the 5% tax on the discounted total:Tax amount = 5% of 1696.50 = 0.05 * 1696.50 = 84.825Round the tax amount to the nearest cent:Tax amount (rounded) = 84.83Add the tax to the discounted total to find the final amount paid:Final amount paid = 1696.50 + 84.83 = 1781.33

💡Okay, so I need to find the sum of the x-intercept and the two y-intercepts of the parabola given by the equation ( x = 3y^2 - 9y + 5 ). Hmm, let me break this down step by step.First, I remember that an x-intercept occurs where the graph crosses the x-axis. At this point, the y-coordinate is 0. So, to find the x-intercept, I should substitute ( y = 0 ) into the equation and solve for x.Let me do that:( x = 3(0)^2 - 9(0) + 5 )Calculating each term:- ( 3(0)^2 = 0 )- ( -9(0) = 0 )- So, ( x = 0 + 0 + 5 = 5 )Therefore, the x-intercept is at the point ( (5, 0) ), which means ( a = 5 ).Next, I need to find the y-intercepts. Y-intercepts occur where the graph crosses the y-axis, which is when the x-coordinate is 0. So, I'll set ( x = 0 ) in the equation and solve for y.Setting ( x = 0 ):( 0 = 3y^2 - 9y + 5 )Now, this is a quadratic equation in terms of y. The standard form of a quadratic equation is ( ay^2 + by + c = 0 ), so here, ( a = 3 ), ( b = -9 ), and ( c = 5 ).To solve for y, I can use the quadratic formula:( y = frac{-b pm sqrt{b^2 - 4ac}}{2a} )Plugging in the values:( y = frac{-(-9) pm sqrt{(-9)^2 - 4(3)(5)}}{2(3)} )Simplify step by step:First, calculate the discriminant (( b^2 - 4ac )):( (-9)^2 = 81 )( 4ac = 4 * 3 * 5 = 60 )So, the discriminant is ( 81 - 60 = 21 ).Now, plug this back into the formula:( y = frac{9 pm sqrt{21}}{6} )So, the two solutions are:( y = frac{9 + sqrt{21}}{6} ) and ( y = frac{9 - sqrt{21}}{6} )Therefore, the y-intercepts are at the points ( (0, frac{9 + sqrt{21}}{6}) ) and ( (0, frac{9 - sqrt{21}}{6}) ), meaning ( b = frac{9 + sqrt{21}}{6} ) and ( c = frac{9 - sqrt{21}}{6} ).Now, I need to find ( a + b + c ). I already have ( a = 5 ), ( b = frac{9 + sqrt{21}}{6} ), and ( c = frac{9 - sqrt{21}}{6} ).Let me add them together:( a + b + c = 5 + frac{9 + sqrt{21}}{6} + frac{9 - sqrt{21}}{6} )I notice that when I add ( frac{9 + sqrt{21}}{6} ) and ( frac{9 - sqrt{21}}{6} ), the ( sqrt{21} ) terms will cancel each other out.Let me compute that:( frac{9 + sqrt{21}}{6} + frac{9 - sqrt{21}}{6} = frac{9 + sqrt{21} + 9 - sqrt{21}}{6} = frac{18}{6} = 3 )So, ( a + b + c = 5 + 3 = 8 ).Wait, that seems straightforward. Let me double-check my steps to make sure I didn't make a mistake.1. Finding the x-intercept by setting ( y = 0 ) gave me ( x = 5 ). That seems correct.2. For the y-intercepts, setting ( x = 0 ) led me to the quadratic equation ( 3y^2 - 9y + 5 = 0 ). Using the quadratic formula, I found the solutions ( y = frac{9 pm sqrt{21}}{6} ). That looks right.3. Adding ( a ), ( b ), and ( c ) together: ( 5 + frac{9 + sqrt{21}}{6} + frac{9 - sqrt{21}}{6} ). The radicals cancel, leaving ( 5 + 3 = 8 ). That makes sense.I think I did everything correctly. So, the sum ( a + b + c ) is 8.**Final Answer**The value of ( a + b + c ) is boxed{8}.

💡Okay, so I have this problem here: Given ( f(x) = x^{alpha} ), where ( alpha ) is a rational number, and it's given that ( f'(-1) = -4 ). I need to find the value of ( alpha ). Hmm, let's see how to approach this.First, I remember that to find the derivative of a function like ( x^{alpha} ), I can use the power rule. The power rule says that if ( f(x) = x^n ), then the derivative ( f'(x) = n x^{n-1} ). So, applying that here, the derivative of ( f(x) = x^{alpha} ) should be ( f'(x) = alpha x^{alpha - 1} ). That seems straightforward.Now, the problem gives me ( f'(-1) = -4 ). That means when I plug ( x = -1 ) into the derivative, the result should be -4. So, let's write that out:( f'(-1) = alpha (-1)^{alpha - 1} = -4 )Okay, so I have the equation ( alpha (-1)^{alpha - 1} = -4 ). I need to solve for ( alpha ). Hmm, this looks a bit tricky because ( alpha ) is both multiplied by and in the exponent with a negative base. I need to figure out how to handle this.Let me think about the properties of exponents with negative bases. If ( (-1)^n ) is involved, the result depends on whether ( n ) is even or odd. If ( n ) is even, ( (-1)^n = 1 ), and if ( n ) is odd, ( (-1)^n = -1 ). But in this case, the exponent is ( alpha - 1 ), which is not necessarily an integer because ( alpha ) is a rational number. Wait, but ( alpha ) is rational, so ( alpha - 1 ) is also rational. Hmm, how does that affect ( (-1)^{alpha - 1} )?I recall that for rational exponents, ( (-1)^{p/q} ) is defined only if ( q ) is odd, because otherwise, you'd be taking an even root of a negative number, which isn't real. So, ( (-1)^{alpha - 1} ) must be a real number, which implies that the denominator of ( alpha - 1 ) when expressed in lowest terms must be odd. That's a bit of a constraint, but maybe it will help later.Let me try to simplify the equation ( alpha (-1)^{alpha - 1} = -4 ). Maybe I can express ( (-1)^{alpha - 1} ) in terms of ( (-1)^{alpha} ) or something similar. Let's see:( (-1)^{alpha - 1} = (-1)^{alpha} times (-1)^{-1} = (-1)^{alpha} times (-1) = -(-1)^{alpha} )So, substituting back into the equation:( alpha times (-(-1)^{alpha}) = -4 )Which simplifies to:( -alpha (-1)^{alpha} = -4 )Multiplying both sides by -1:( alpha (-1)^{alpha} = 4 )Hmm, that's interesting. So now I have ( alpha (-1)^{alpha} = 4 ). I need to find a rational number ( alpha ) such that when I multiply it by ( (-1)^{alpha} ), I get 4.Let me think about possible values of ( alpha ). Since ( alpha ) is rational, let's assume it's a fraction ( frac{p}{q} ) where ( p ) and ( q ) are integers with no common factors, and ( q ) is positive.So, ( (-1)^{alpha} = (-1)^{p/q} ). As I thought earlier, for this to be real, ( q ) must be odd. So, ( q ) is an odd integer.Now, ( (-1)^{p/q} ) can be either 1 or -1 depending on whether ( p ) is even or odd. Wait, no, that's not quite right. Actually, ( (-1)^{p/q} ) is equal to ( cos(pi p/q) + i sin(pi p/q) ), but since we're dealing with real numbers, ( (-1)^{p/q} ) is real only if ( p/q ) is an integer or if ( q ) is odd. But in this case, since ( q ) is odd, ( (-1)^{p/q} ) is real and equal to ( (-1)^p ), because ( (-1)^{1/q} ) is -1 if ( q ) is odd, right? Wait, no, that's not correct.Actually, ( (-1)^{1/q} ) where ( q ) is odd is equal to -1, because ( (-1)^q = -1 ), so the q-th root of -1 is -1. Therefore, ( (-1)^{p/q} = [(-1)^{1/q}]^p = (-1)^p ). So, ( (-1)^{p/q} = (-1)^p ).Therefore, ( (-1)^{alpha} = (-1)^p ). So, ( (-1)^{alpha} ) is either 1 or -1, depending on whether ( p ) is even or odd.So, going back to the equation ( alpha (-1)^{alpha} = 4 ), we have two cases:Case 1: ( (-1)^{alpha} = 1 ). Then, ( alpha times 1 = 4 ), so ( alpha = 4 ).Case 2: ( (-1)^{alpha} = -1 ). Then, ( alpha times (-1) = 4 ), so ( alpha = -4 ).But wait, I need to check if these values of ( alpha ) satisfy the original equation ( f'(-1) = -4 ).Let's test ( alpha = 4 ):( f'(x) = 4 x^{3} )So, ( f'(-1) = 4 (-1)^3 = 4 (-1) = -4 ). That works.Now, let's test ( alpha = -4 ):( f'(x) = -4 x^{-5} )So, ( f'(-1) = -4 (-1)^{-5} = -4 times (-1)^{-5} ). Now, ( (-1)^{-5} = 1/(-1)^5 = 1/(-1) = -1 ). So, ( f'(-1) = -4 times (-1) = 4 ). But we need ( f'(-1) = -4 ), so this doesn't work.Therefore, ( alpha = -4 ) is not a valid solution. So, the only solution is ( alpha = 4 ).Wait, but I need to make sure that ( alpha = 4 ) is indeed a rational number, which it is, since 4 is an integer and integers are rational numbers.Also, I should check if there are any other possible values of ( alpha ) that could satisfy the equation. For example, could there be a fractional ( alpha ) that also works?Let me consider ( alpha = frac{p}{q} ), where ( p ) and ( q ) are integers with ( q ) odd.From the equation ( alpha (-1)^{alpha} = 4 ), and knowing that ( (-1)^{alpha} = (-1)^p ), we have:( frac{p}{q} times (-1)^p = 4 )So, ( frac{p}{q} times (-1)^p = 4 )Since ( frac{p}{q} ) is a rational number and ( (-1)^p ) is either 1 or -1, the product must be 4. So, ( frac{p}{q} ) must be either 4 or -4.But ( frac{p}{q} ) is in lowest terms, so ( p ) and ( q ) have no common factors. If ( frac{p}{q} = 4 ), then ( p = 4q ). But since ( p ) and ( q ) have no common factors, ( q ) must be 1. Therefore, ( alpha = 4 ).Similarly, if ( frac{p}{q} = -4 ), then ( p = -4q ). Again, since ( p ) and ( q ) have no common factors, ( q ) must be 1, so ( alpha = -4 ). But as we saw earlier, ( alpha = -4 ) doesn't satisfy the original equation because it gives ( f'(-1) = 4 ) instead of -4.Therefore, the only valid solution is ( alpha = 4 ).I think that's thorough enough. I considered both cases where ( (-1)^{alpha} ) is 1 or -1, checked the solutions, and ensured that there are no other possible rational values for ( alpha ) that would satisfy the equation. So, I'm confident that ( alpha = 4 ) is the correct answer.