
Math
We wish to evaluate the expression `10^(((x+y)/2)  3z)` First, simplify the power into seperate terms involving each of `x`, `y` and `z` only: `10^(((x+y)/2)  3z) = 10^(x/2 + y/2  3z)` Then...

Math
First evaluate the expression under the square root sign: `x^2 + 2x + (y1)^2  2xy = x^2 + 2x + y^2  2y + 1  2xy` `= x^2 + y^2 2xy + 2(xy) + 1` Now `(xy)^2 = x^2 + y^2  2xy` So that...

Math
a) We want the roots of `x^3 +3x^2 4x + d=0` If we are given that two of the roots are opposites then we have that `x^3 + 3x^2  4x + d = (xa)(x+a)(xb)` `= (x^2 a^2)(xb)` Multiply this out...

algebra1
y3=3(x+1)If we are going to express this in slopeintercept form y=mx+b, then we have todistribute 3 to x+1.y3=3x+3And, isolate the y by adding 3 on both sides of the...

algebra1
the answer is NOT 5! C(a,b) may also be written as aCb, implying a!/b!(ab)! Hence, (n+1)C3 = (n+1)!/3!(n+13)! But (n+1)! = (n+1)n! And (n+13)! = (n2)! Therefore, (n+1)C3= (n+1)n!/3!(n2)! Also,...

algebra1
We have to prove that a^2  4a + b^2 + 10b + 29>=0, for real values of a and b. a^2  4a + b^2 + 10b + 29 => a^2  4a + 4 + b^2 + 10b + 25 => (a  2)^2 + (b + 5)^2 The sum of squares of...

algebra1
k! = 1*2*3*...*k (k  1)/k! = k/k!  1/k! => 1/(k  1)!  1/k! The sum of (k  1)/k! for k = 1 to n is: 1/0!  1/1! + 1/1!  1/2! + 1/2!  1/3! + ... + 1/(n  1)!  1/n! => 1/0!  1/n!...

algebra1
We'll apply the quotient rule of logarithms: log (a/b) = log a  log b According to this rule, we'll get: log (2x5)/(x^2+3) = log (2x5)  log(x^2+3) We'll rewrite the equation: log (2x5) ...

algebra1
We'll replace the result of the difference x  y by z, at the numerator of the 1st option. (xy)/2z = z/2z We'll simplify and we'll get: (xy)/2z = 1/2 Since the result is not equal 2, we'll reject...

algebra1
The absolute value of the complex number can be evaluated when we know the rectangular form of z: z = x + i*y z = sqrt(x^2 + y^2) We'll identify the real part and the imaginary part of z: x =...

algebra1
We'll multiply the first fraction by (3x  5) and the 2nd by (4x1)[3x(3x  5) + 2x(4x1)]/(4x1)(3x5)We'll remove the brackets:(9x^2  15x + 8x^2  2x)/(12x^2  20x  3x + 5)(17x^2  17x)/(12x^2...

algebra1
To determine the expression of f(x^2), we'll simply replace x by x^2 in the expression of the function: f(x^2) = 4*(x^2)^2  3 f(x^2) = 4x^4  3 The found expression of f(x^2) is f(x^2) = 4x^4  3

algebra1
The linear function put in standard form is: f(x) = ax + b Since the graph of the function is passing through the points (1,2) and (3,1), that means that if we'll substitute the coordinates of the...

algebra1
We'll apply quadratic formula to determine the roots: b1 = [(21)+sqrt((21)^2 + 4*108)]/2*1 b1 = (21+sqrt9)/2 b1 = (21+3)/2 b1 = 12 b2 = (213)/2 b2 = 9 We can write the quadratic expression as a...

algebra1
We'll rewrite g as a product of linear factors, using it's roots. x1 = [1 + sqrt(1 + 8)]/2 x1 = (1+3)/2 x1 = 2 x2 = (13)/2 x2 = 1 Therefore, g = (x2)(x+1) Since f is divisible by g, it means...

algebra1
We'll determine the inverse of g(x) in this way. Let g(x) = y y = 2/(x+1) Now, we'll find x with respect to y. F y(x+1) = 2 We'll remove the brackets and we'll get: yx + y = 2 We'll isolate x to...

algebra1
We'll shift 6 to the left side: 5126+6 =  5(x+ 22)^(5/3) 5120 = 5(x+ 22)^(5/3) We'll divide by 5 both sides: 1024 = (x+ 22)^(5/3) We'll raise both sides to 3/5 power: 1024^(3/5) = x+22 We'll...

algebra1
Writing 9 as a power of 3, we'll create matching bases both sides: 9 = 3^2 We'll apply the rule of negative power: a^b = 1/a^b We'll put a = 9 and b = 3 9^3 = 1/9^3 = 1/(3^2)^3 = 1/3^6 = 3^6...

algebra1
We'll choose to rewrite f(x): ( 2x^3 + 2x  1 )/ (x^2 + 1) = (2x^3 + 2x)/(x^2 + 1)  1/(x^2 + 1) We'll factorize by 2x the first fraction: (2x^3 + 2x)/(x^2 + 1) = 2x(x^2 + 1)/(x^2 + 1) We'll...

algebra1
We'll write the rectangular form of any complex number is z = x + y*i. The trigonometric form of a complex number is: z = z(cos a + i*sin a) z = sqrt(x^2 + y^2) cos a = x/z sin a = y/z...

algebra1
We'll ilustrate the idea with the help of the following example: 3*2^2x  5*2^x*3^x + 2*3^2x= 0 We'll divide by 3^2x to create matching bases that can be replaced by another variable, later on:...

algebra1
Since the composition of 2 functions is not commutative, then (fog)(x) is not the same with (gof)(x) We'll get (fog)(x) for f(x) = x^2 and g(x) = (x7). (fog)(x) = f(g(x)) We'll substitute x by...

algebra1
We'll start by imposing the constraint of existence of the square roots: x+2 >=0 x> =2 2x>=0 x=<2 The interval of possible values for x is [2 ; 2]. Now, we'll solve the equation....

algebra1
We notice that 6^x = (2*3)^x But (2*3)^x = 2^x*3^x We'll subtract 5*2^x*3^x both sides: 3*2^2x + 2*3^2x  5*2^x*3^x = 0 We'll divide by 3^2x: 3*(2/3)^2x  5*(2/3)^x + 2 = 0 We'll note (2/3)^x = t...

algebra1
We'll use the theorem of arithmetic mean to determine the terms of the arithmetic progression. 4 = (x+y)/2 => 8 = x+y (1) y = (4+12)/2 y = 16/2 y = 8 We'll substitute y into (1): 8 = x+8 We'll...

algebra1
To find the local extreme of a function, first we have to determine the critical value of the function. The critical value of the function is the root of the first derivative of the function....

algebra1
Before solving the equation, we'll impose conditions of existence of the square root. 5x6 >= 0 We'll subtract 6 both sides: 5x >= 6 We'll divide by 5: x >=6/5 The interval of admissible...

algebra1
We'll impose the constraints of existence of logarithm. x>0 The solution has to be in the interval of admissible values (0,+infinite) lgx/(1lg2) = 2 lgx = 2  2*lg2 Well use the power rule of...

algebra1
We'll denote the point with that has equal coordinates as M(m,m). Since the point is located on the line y = 0.5x  0.5, it's coordinates verify the expression of the line. We'll put y = f(x) and...

algebra1
The complex roots of a polynomial are always found as conjugate pairs. For the root 2  i, the complex conjugate is 2 + i. The polynomial has the roots 2  i and 2 + i Irrational roots are also...

algebra1
We'll establish the constraints of existence of logarithms: x  24/5>0 x > 24/5 2x>0 x>0 The range of admissible values for x is (24/5 ,+inf.). We'll apply quotient property of...

algebra1
The function f(x) = x^2  tx  3. The point (2, 9) lies on the curve. => 9 = 2^2  t*2  3 => 9 = 4  2t  3 => 8 = 2t => t = 4 The coefficient t = 4

algebra1
We have the points P(7,11) and Q(2,4), and we need to find the length, slope and the midpoint of the line segment joining them. The length of the line segment is: sqrt ((7 + 2)^2 + (11  4)^2)...

algebra1
We have to simplify: 4t^2 16/8/t  2/6 4t^2 16/8/t  2/6 => 4t^2 (16/8)/t  (2/6) 16/8 = 2 and 2/6 = 1/3 => 4t^2 2/t  1/3 We can simplify 4t^2 16/8/t  2/6 as 4t^2 2/t  1/3.

algebra1
We have to solve 13/207/10x=0.5 for x 13/207/10x=0.5 => 13x/20x  14/20x = 1/2 => (13x  14)/20x = 1/2 => 13x  14 = 10x => 3x = 14 => x = 14/3 The value of x = 14/3

algebra1
To solve 2x  y = 5 ...(1) 3x  2y = 9 ...(2) using substitution take (1) 2x  y = 5 => y = 2x  5 substitute in (2) 3x  2(2x  5) = 9 => 3x  4x + 10 = 9 => x = 1 => x = 1 y = 2x ...

algebra1
A polynomial has complex roots in pairs of conjugates. As the polynomial has roots 2 and 2i, it also has 2i as a root. The polynomial is: (x  2)(x  2i)(x + 2i) => (x  2)(x^2  4i^2) => (x...

algebra1
Consecutive terms of a GP have a common ratio. if 2, x, y, 16 form a GP. => 16/y = x/2 => x = 32/y y/x = x/2 Substitute x = 32/y => y/(32/y) = (32/y)/2 => 2y = (32/y)^2 => 2y^3 =...

algebra1
We have to solve x^4  3x^2 + 2 = 0 x^4  3x^2 + 2 = 0 => x^4  2x^2  x^2 + 2 = 0 => x^2(x^2  2)  1(x^2  2) = 0 => (x^2  1)(x^2  2) = 0 x^2 = 1 => x = 1 , x = 1 x^2 = 2 => x =...

algebra1
We have to find the complex number z given that (3z  2z')/6 = 5 Let z = a + ib, z' = a  ib (3z  2z')/6 = 5 =>(3(a + ib)  2(a  ib)) = 30 => 3a + 3ib  2a + 2ib = 30 => a + 5ib =...

algebra1
We'll write the given expresison as a fraction, using the negative power property: (7xx^2)^1 = 1/(7xx^2) We'll get 2 elementary fractions because we notice 2 factors at denominator. 1/(7xx^2)...

algebra1
We'll remove the brackets, using FOIL method: (2i+5)(i+7) = 2i^2  14i + 5i + 35 We know that i^2 = 1 (2i+5)(i+7) = 2  9i + 35 We'll combine real parts: (2i+5)(i+7) = 37  9i The result of...

algebra1
It is given that 3z 9i = 8i + z + 4 3z 9i = 8i + z + 4 => 3z  z = 4 + 8i + 9i => 2z = 4 + 17i => z = 2 + 17i/2 z = sqrt (2^2 + (17/2)^2) => sqrt (4 + 289/4) => sqrt (305/4)...

algebra1
It is given that log(72) 48 = a and log(6) 24 = b a = log(72) 48 = log(6) 48/ log(6) 72 => log(6) (6*8)/log(6) 6*12 => [1 + log(6) 8]/[1 + log(6) 12] => [1 + 3*log(6) 2]/[2 + log(6) 2] b...

algebra1
It is given that log x^3 log 10x = log 10^5. To determine x , use the property : log a  log b = log a/b log x^3 log 10x = log 10^5 => log (x^3 / 10x) = 10^5 => x^2 / 10 = 10^5 => x^2 =...

algebra1
The equation to be solved is : (3x+7)(x1) = 24 (3x+7)(x1) = 24 => 3x^2 + 4x  7 = 24 => 3x^2 + 4x  31 = 0 x1 = 4/6 + sqrt (16 + 372) /6 => 2/3 + (sqrt 97)/6 x2 = 2/3  (sqrt 97)/6...

algebra1
We have to solve 3^(3x9) = 1/81 for x 3^(3x9) = 1/81 => 3^(3x9) = 3^(4) as the base is the same equate the exponent 3x  9 = 4 => 3x = 5 => x = 5/3 The solution is 5/3

algebra1
The equation to be solved is (x4)^1/2=1/(x4) (x4)^1/2=1/(x4) square both the sides => x  4 = 1 / (x  4)^2 => (x  4)^3 = 1 => 1  (x  4)^3 = 0 => (1  (x  4))(1 + x  4 + (x ...

algebra1
We need to solve (log(2) x)^2 + log(2) (4x) = 4 Use the property that log a*b = log a + log b (log(2) x)^2 + log(2) (4x) = 4 => (log(2) x)^2 + log(2) 4 + log(2) x = 4 => (log(2) x)^2 + 2 +...

algebra1
The value of x if 4^(4x  15)  4 = 0 can be found by rewriting the equation as 4^(4x  15) = 4 Now, as the base is the same on both the sides, equate the exponent 4x  15 = 1 => 4x = 16 => x...