💡Okay, so I have this quadratic equation: ( x^2 - x + a = 0 ). The question is asking about the possible values of ( a ) that would make this equation have real roots. The options given are A: 2, B: 5, C: 0.5, and D: 0.25. Hmm, I need to figure out which of these values of ( a ) will ensure that the equation has real roots.First, I remember that for a quadratic equation of the form ( ax^2 + bx + c = 0 ), the discriminant determines the nature of the roots. The discriminant is given by ( D = b^2 - 4ac ). If the discriminant is positive, the equation has two distinct real roots. If it's zero, there's exactly one real root (a repeated root), and if it's negative, the roots are complex and not real.In this case, the equation is ( x^2 - x + a = 0 ). Comparing this to the standard form ( ax^2 + bx + c = 0 ), I can identify the coefficients:- ( a = 1 ) (the coefficient of ( x^2 ))- ( b = -1 ) (the coefficient of ( x ))- ( c = a ) (the constant term, which is given as ( a ) in the problem)So, plugging these into the discriminant formula:[ D = (-1)^2 - 4(1)(a) ][ D = 1 - 4a ]For the equation to have real roots, the discriminant must be greater than or equal to zero:[ 1 - 4a geq 0 ][ 1 geq 4a ][ a leq frac{1}{4} ]So, ( a ) must be less than or equal to ( frac{1}{4} ) or 0.25. Now, looking at the options provided:- A: 2 (which is greater than 0.25)- B: 5 (also greater than 0.25)- C: 0.5 (still greater than 0.25)- D: 0.25 (exactly equal to 0.25)Therefore, only option D satisfies the condition ( a leq 0.25 ). So, the value of ( a ) can be 0.25.Wait, just to make sure I didn't make a mistake. Let me double-check my steps:1. Identified the coefficients correctly: ( a = 1 ), ( b = -1 ), ( c = a ).2. Calculated the discriminant: ( D = (-1)^2 - 4(1)(a) = 1 - 4a ).3. Set discriminant greater than or equal to zero: ( 1 - 4a geq 0 ).4. Solved for ( a ): ( a leq frac{1}{4} ).5. Compared the options: Only D is 0.25, which is equal to ( frac{1}{4} ).Yep, that seems right. So, I'm confident that D is the correct answer.

💡First, identify the coordinates of points M and N.Point M has coordinates (1, 2, 3), and point N has coordinates (2, 3, 4).Next, apply the distance formula for two points in three-dimensional space:[text{Distance} = sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}]Substitute the coordinates into the formula:[sqrt{(2 - 1)^2 + (3 - 2)^2 + (4 - 3)^2}]Calculate each difference:[(2 - 1) = 1,quad (3 - 2) = 1,quad (4 - 3) = 1]Square each difference:[1^2 = 1,quad 1^2 = 1,quad 1^2 = 1]Sum the squared differences:[1 + 1 + 1 = 3]Take the square root of the sum:[sqrt{3}]Thus, the distance between points M and N is (sqrt{3}).

💡Okay, so I have this problem about an arithmetic sequence and some vectors. Let me try to understand what it's asking.First, the sequence {a_n} is an arithmetic sequence. That means each term increases by a constant difference. So, a_n = a_1 + (n-1)d, where d is the common difference.The problem gives me a vector equation: BA = a_3 OB + a_{2015} OC. Hmm, vectors. I need to recall some vector properties. BA is the vector from point B to point A, right? So, BA = OA - OB, where OA and OB are vectors from the origin to points A and B respectively.So, substituting that into the equation: OA - OB = a_3 OB + a_{2015} OC. Let me write that down:OA - OB = a_3 OB + a_{2015} OCThen, if I move OB to the right side:OA = (1 + a_3) OB + a_{2015} OCOkay, so OA is expressed in terms of OB and OC. Now, the problem also says that AB = λ AC, where λ is a real number. AB is the vector from A to B, which is OB - OA. Similarly, AC is the vector from A to C, which is OC - OA.So, AB = λ AC translates to:OB - OA = λ (OC - OA)Let me substitute OA from the earlier equation into this:OB - [(1 + a_3) OB + a_{2015} OC] = λ [OC - (1 + a_3) OB - a_{2015} OC]Simplify the left side:OB - (1 + a_3) OB - a_{2015} OC = (1 - 1 - a_3) OB - a_{2015} OC = (-a_3) OB - a_{2015} OCNow, the right side:λ [OC - (1 + a_3) OB - a_{2015} OC] = λ [(-1 - a_3) OB + (1 - a_{2015}) OC]So, putting it all together:(-a_3) OB - a_{2015} OC = λ [(-1 - a_3) OB + (1 - a_{2015}) OC]Since vectors OB and OC are linearly independent (because point O is outside line BC), their coefficients must be equal on both sides. So, we can set up equations for the coefficients:For OB:-a_3 = λ (-1 - a_3)For OC:-a_{2015} = λ (1 - a_{2015})So, now I have two equations:1) -a_3 = λ (-1 - a_3)2) -a_{2015} = λ (1 - a_{2015})Let me solve equation 1 for λ:From equation 1:-a_3 = -λ (1 + a_3)Divide both sides by -1:a_3 = λ (1 + a_3)So, λ = a_3 / (1 + a_3), provided that 1 + a_3 ≠ 0.Similarly, from equation 2:-a_{2015} = λ (1 - a_{2015})So, λ = (-a_{2015}) / (1 - a_{2015}), provided that 1 - a_{2015} ≠ 0.Since both expressions equal λ, we can set them equal to each other:a_3 / (1 + a_3) = (-a_{2015}) / (1 - a_{2015})Cross-multiplying:a_3 (1 - a_{2015}) = -a_{2015} (1 + a_3)Let me expand both sides:Left side: a_3 - a_3 a_{2015}Right side: -a_{2015} - a_3 a_{2015}So, bringing all terms to one side:a_3 - a_3 a_{2015} + a_{2015} + a_3 a_{2015} = 0Simplify:a_3 + a_{2015} = 0So, a_3 + a_{2015} = 0But since {a_n} is an arithmetic sequence, we know that a_n = a_1 + (n - 1)d.So, a_3 = a_1 + 2da_{2015} = a_1 + 2014dAdding them together:a_3 + a_{2015} = (a_1 + 2d) + (a_1 + 2014d) = 2a_1 + 2016d = 0So, 2a_1 + 2016d = 0Divide both sides by 2:a_1 + 1008d = 0Hmm, interesting. So, a_1 = -1008dNow, the question asks for a_1 + a_{2017}Let me compute a_{2017}:a_{2017} = a_1 + 2016dSo, a_1 + a_{2017} = a_1 + (a_1 + 2016d) = 2a_1 + 2016dBut from earlier, we have 2a_1 + 2016d = 0Therefore, a_1 + a_{2017} = 0So, the answer is 0, which is option A.Wait, let me double-check my steps to make sure I didn't make a mistake.Starting from the vector equation, I expressed OA in terms of OB and OC. Then, using AB = λ AC, I substituted OA into that equation and equated coefficients. That led me to two equations for λ, which I set equal to each other. After some algebra, I found that a_3 + a_{2015} = 0. Then, using the arithmetic sequence formula, I expressed a_3 and a_{2015} in terms of a_1 and d, leading to 2a_1 + 2016d = 0. Then, since a_1 + a_{2017} is 2a_1 + 2016d, which equals 0. So, yeah, that seems correct.I think I did everything right. So, the answer should be 0.**Final Answer**boxed{A}

💡Let's solve the problem step by step.1. **Assign Variables:** - Let Walter's age at the end of 1994 be ( x ). - Since Walter was half as old as his grandmother at the end of 1994, his grandmother's age at that time would be ( 2x ).2. **Determine Birth Years:** - Walter was born in the year ( 1994 - x ). - His grandmother was born in the year ( 1994 - 2x ).3. **Set Up the Equation:** - The sum of their birth years is given as 3838. Therefore: [ (1994 - x) + (1994 - 2x) = 3838 ]4. **Simplify the Equation:** [ 1994 - x + 1994 - 2x = 3838 3988 - 3x = 3838 -3x = 3838 - 3988 -3x = -150 x = frac{-150}{-3} x = 50 ] So, Walter was 50 years old at the end of 1994.5. **Calculate Walter's Age in 1999:** - From 1994 to 1999 is 5 years. - Walter's age at the end of 1999: [ 50 + 5 = 55 ]6. **Conclusion:** - Walter will be 55 years old at the end of 1999.(boxed{D})

💡First, I need to determine the total number of towel sets Bailey is purchasing. She is buying 2 sets for the guest bathroom and 4 sets for the master bathroom, which totals 6 sets.Next, I'll calculate the cost of the guest bathroom towels. Each set costs 40.00, so for 2 sets:2 sets × 40.00/set = 80.00Then, I'll calculate the cost of the master bathroom towels. Each set costs 50.00, so for 4 sets:4 sets × 50.00/set = 200.00Now, I'll add the costs of both sets to find the total cost before the discount:80.00 (guest) + 200.00 (master) = 280.00The store offers a 20% discount on the total purchase. To find the discount amount:20% of 280.00 = 0.20 × 280.00 = 56.00Finally, I'll subtract the discount from the total cost to find the final amount Bailey will spend:280.00 - 56.00 = 224.00Bailey will spend 224.00 on towel sets after applying the 20% discount.

💡First, I'll denote the initial volumes of liquids ( p ) and ( q ) based on the given ratio of ( 3:2 ). Let the initial volume of liquid ( p ) be ( 3x ) liters and the initial volume of liquid ( q ) be ( 2x ) liters, where ( x ) is a common factor.The initial total volume of the mixture is the sum of the volumes of liquids ( p ) and ( q ):[V_{text{initial}} = 3x + 2x = 5x]Next, we are told that adding 1 liter of liquid ( q ) changes the ratio to ( 4:3 ). After adding 1 liter, the volume of liquid ( q ) becomes ( 2x + 1 ) liters. The new ratio of liquid ( p ) to liquid ( q ) is ( 4:3 ), so we can set up the following equation:[frac{3x}{2x + 1} = frac{4}{3}]Cross-multiplying to solve for ( x ):[9x = 4(2x + 1)][9x = 8x + 4][9x - 8x = 4][x = 4]Now that we have the value of ( x ), we can find the initial volume of the mixture:[V_{text{initial}} = 5x = 5 times 4 = 20 text{ liters}]