## algebra1 Questions and Answers

algebra1

### Solve for x log2 x + log4 x + log8 x =11/6

We have to solve for x: log2 x + log4 x + log8 x =11/6 log (a) b = 1/ log (b) a log2 x + log4 x + log8 x =11/6 => 1 / log(x) 2 + 1 / log(x) 4 + 1/log(x) 2 = 11/6 => 1 / log(x) 2 + 1 /...

algebra1

### Evaluate the summation (k-1)/k! if k goes from 1 to n.

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

### Find the product of roots of quadratic equation if |x1-x2|=1 and x^2=2x-m.

The roots of the quadratic equation are x1 and x2. |x1 - x2| = 1 and x^2 = 2x - m As x^2 = 2x - m x1^2 = 2* x1 - m x2^2 = 2*x2 - m Subtracting the two we get x1^2 - x2^2 = 2( x1 - x2) => (x1 -...

algebra1

### Solve the system of equations algebraically x^2+y^2=100 x-y=2

We have x^2 + y^2 = 100 and x - y = 2. We have to solve these equations for x and y. x - y = 2 => (x - y)^2 =2^2 => x^2 + y^2 - 2xy = 4 substitute x^2 + y^2 = 100 => 100 - 2xy = 4 =>...

algebra1

### Find the first term and the common difference of the arithmetic sequence if a2-a6+a4=-7 and a8-a7=2a4

The nth term of an arithmetic sequence can be written as a + (n - 1)*d, where a is the first term and d is the common difference. We have a2-a6+a4=-7 => a + d - a - 5d + a + 3d = -7 => a - d...

algebra1

### Inverse of a number: find the multiplicative inverse of the number 3 + 2i.

The multiplicative inverse is another name for the reciprocal. Both numbers are such that, if multiplied by the given number, they result in 1. The number 1 is special with regard to the operation...

algebra1

### If the roots of the equation x^3-9x^2+23x-15=0,are in AP,then one of its roots will be (1)3,(2)9,(3)15(4)0

The roots of x^3 - 9x^2 + 23x - 15 = 0 are in AP. x^3 - 9x^2 + 23x - 15 = 0 => x^3 - 3x^2 - 6x^2 + 18x + 5x - 15 = 0 => x^2(x -3 ) - 6x ( x - 3) + 5(x - 3) =0 => (x^2 - 6x + 5)(x - 3) = 0...

algebra1

### logarithmsWhat is x if (log2 x)^2+log2 (4x)=4?

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

### Given the roots 6 and 7, determine the quadratic equation?

The roots of a quadratic equation are given as 6 and 7. This means that x - 6 = 0 and x - 7 = 0 We can write ( x - 6)( x - 7) = 0 => x^2 - 6x - 7x + 42 = 0 => x^2 - 13x + 42 = 0 Therefore the...

algebra1

### Solve for x and y x^3-y^3=7 x^2+xy+y^2=7

x^3 - y^3 = 7..........(1) x^2 + xy + y^2 = 7 ..............(2) We need to solve the system. First we will rewrite equation (1) as a difference between cubes. ==> x^3 - y^2 = (x-y)(x^2 +...

algebra1

### Find all the zeros of the function f(x)=x^2-8x-9

We have to find the zeros of the function f(x) = x^2 - 8x - 9 For this we equate f(x) to 0 and solve the quadratic equation that is obtained. f(x) = 0 => x^2 - 8x - 9 = 0 => x^2 - 9x + x - 9...

algebra1

### Solve the following simultaneous equations. 2x – 3y = 5 , x – 2y = 4

We have to solve the following set of simultaneous equations: 2x - 3y = 5 ...(1) x - 2y = 4 ...(2) From (2) x - 2y = 4 => x= 4 + 2y Substitute in (1) 2( 4 + 2y) - 3y = 5 => 8 + 4y - 3y = 5...

algebra1

### square root(x+square root(1-x))+square rootx=1. What is x?

We have to solve the equation sqrt (x + sqrt (1 - x)) + sqrt x = 1 sqrt (x + sqrt (1 - x)) + sqrt x = 1 sqrt (x + sqrt (1 - x)) = 1 - sqrt x square both the sides => x + sqrt(1 - x) = 1 + x - 2...

algebra1

### Determine the imaginary part of complex number z if z^2=3+4i .

We have to determine the imaginary part of the complex number z for z^2 = 3 + 4i Let z = x + yi z^2 = x^2 + y^2*i^2 + 2xyi = 3+ 4i => x^2 - y^2 + 2xyi = 3 + 4i equate the real and imaginary...

algebra1

### What is the multiplicative inverse of 6-3i?

The multiplicative inverse of an element of a set is another element of this set such that the product of the two elements is a multiplicative identity. In the given set of complex numbers, the...

algebra1

### Write the polar form of the complex number z given by z=6-8i.

The polar form of a complex number z = x + yi is r*(cos A + i* sin A) where tan A= y/x and r = sqrt (x^2 + y^2) Here we have z = 6 - 8i r = sqrt( 6^2 + 8^2) = sqrt (36 + 64) = sqrt 100 = 10 tan A =...

algebra1

### What is the first term of arithmetic sequence if the sum a1+a2+...a13=130? a4,a10,a7 are also consecutive terms of...

Let the first term of the arithmetic sequence be a and the common difference be d. (a + a + 12d)*(13/2) = 130 => 2a + 12d = 20 => a + 6d = 10 => a = 10 - 6d a4, a10 and a7 are consecutive...

algebra1

### Geometric Progression What is the value of x and y if 2, x, y, 16 form a geometric progression ?

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

### What are x and y if 4^(x/y)*4^(y/x)=32 and log 3 (x-y)=1-log 3 (x+y) ?

In this question, I am not clear whether the logs have the base of 3 or 3 is the part of the arguments. I will try to show both cases. Let's consider the given system of equations: 4^(x/y)*4^(y/x)...

algebra1

### What is the real solution of the system x^2+y^2=16, xy=3 ?

We have the system of equations x^2 + y^2 = 16 and xy = 13 to solve for x. xy = 3 => x = 3/y...(1) Substitute this in x^2 + y^2 = 16 => (3/y)^2 + y^2 = 16 => 9 + y^4 = 16y^2 => y^4 -...

algebra1

### 15^(2x-3)=3^x*5^(3x-6) x=?

We have to solve for x given: 15^(2x-3)=3^x*5^(3x-6) 15^(2x-3)=3^x*5^(3x-6) => (5*3)^(2x - 3) = 3^x * 5^(3x - 6) => 5^(2x - 3) * 3^(2x - 3) = 3^x * 5^(2x - 3)* 5^x / 5^3 => 3^(2x - 3) =...

algebra1

### How to find domain of function f(x)=(x-2)/(x^2-4)?

The domain of a function f(x) is all the values x for which f(x) gives real values. f(x)=(x-2)/(x^2-4) => (x - 2)/(x - 2)(x + 2) => 1/(x + 2) This is not defined when x = -2 The domain is R -...

algebra1

### Given the string (an),n>=1 and the sum a1+a2+a3+...+an=(5n^2+6n), what are an and a1?

We are given that a1 + a2 + a3 + ... + an = (5n^2+6n). Sn = a1 + a2 + a3 + ... + an = (5n^2+6n). Sn+1 = a1 + a2 + a3 + ... + an + an+1 = (5(n+1)^2+6(n+1)). Sn+1 - Sn => a1 + a2 + a3 + ... + an +...

algebra1

### What are the real solutions of the equation log(x+2) x + logx (x+2)=5/2 ?

We have to find the real solutions of log(x+2) x + log(x) (x+2) = 5/2. We use the relation log (a) b = 1/log (b) a and log a + log b = log a*b log(x+2) x + log(x) (x+2) = 5/2 => 1/ log (x) (x+2)...

algebra1

### Solve the equation in z : 2z+6i = z/2i+5i-7.

We have to solve 2z + 6i = z/2i + 5i - 7 2z + 6i = z/2i + 5i - 7 => 2z - z/ 2i = -i - 7 => (2z*2i - z) / 2i = (-i - 7) => z( 4i - 1)/2i = (-i - 7) => z = (-i - 7) * 2i / ( 4i - 1) =>...

algebra1

### Complex numbers Sum the complex numbers 3+2i+1-5i

The sum of 3+2i+1-5i is found by adding the real terms together and the complex terms, or the terms that contain i, together. 3+2i+1-5i => 3 + 1 + 2i - 5i => 4 - 3i The required result of...

algebra1

### Solve for x the equation 5(8e^2x - 3)^3 = 625?

We have to solve: 5(8e^2x - 3)^3 = 625 for x. 5(8e^2x - 3)^3 = 625 divide both sides by 5 => (8e^2x - 3)^3 = 125 take the cube root of both the sides => 8e^2x - 3 = 5 add 3 to both the sides...

algebra1

### Solve the equation. e^2x - 6e^x + 5 = 0

We have to solve: e^2x - 6e^x + 5 = 0 e^2x - 6e^x + 5 = 0 let e^x = t => t^2 - 6t + 5 = 0 => t^2 - 5t - t + 5 = 0 => t(t - 5) - 1( t - 5) = 0 => (t - 1)(t - 5) = 0 => e^x = 1 and e^x...

algebra1

### Find the argument of the complex number (2+2i)^11/(2-2i)^9

The complex number (2+2i)^11/(2-2i)^9 has to be simplified first. (2+2i)^11/(2-2i)^9 => (4 + 4i^2 + 8i)^5*(2 + 2i) / (4 +4i^2 - 8i)^4 * (2 - 2i) => (8i)^5*(2 + 2i) / (- 8i)^4 * (2 - 2i) =>...

algebra1

### Function Determine whether the function f(x)=(2x-5)/(7x+4) has an inverse and , if so , find the inverse.

We see that for each value of x, f(x)=(2x-5)/(7x+4) has only one value and each value of f(x) can be obtained by only one value of x. The function has an inverse. Let y = f(x)=(2x-5)/(7x+4) express...

algebra1

### Solve the simultaneous equations x^2+y^2=10 , x^4+y^4=82

We have to solve the simultaneous equations x^2 + y^2=10 ...(1) x^4 + y^4=82 ...(2) let a = x^2 and b = y^2 This gives a + b = 10 and a^2 + b^2 = 82 a + b = 10 => a^2 + b^2 + 2ab = 100...

algebra1

### What is (f o g)(36) if f(x)=6^x and g(x)=log6 x ?

We are given that f(x)= 6^x and g(x) = log(6) x We have to find fog(36) fog(36) = f(g(36)) => f(log(6) 36) => 6^(log (6) 36) we know that a^(log(a) x) = x => 36 The required solution for...

algebra1

### Polynomials . Find the polynomial with lowest degree having the following zeros : 2 and 2i .

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

### need to calculate sum of 3 consecutive terms of arithmetic sequence x+4,3x,13x-2

The terms x + 4, 3x and 13x - 2 are consecutive terms of an arithmetic progression. So we know that they have a common difference 13x - 2 - 3x = 3x - x - 4 => 10x - 2 = 2x - 4 => 8x = -2...

algebra1

### What is polynomial ax^3+bx^2+cx+d if divided by (x-1),(x+1),(x+2), the reminder is always 3?

The polynomial ax^3 + bx^2 + cx + d is the product of the terms (x-1), (x+1) and (x+2) to which the remainder 3 is added. (x - 1)(x +1)(x + 2) +3 => (x^2 - 1)(x + 2) +3 => x^3 - x + 2x^2 - 2...

algebra1

### x^log2 x + 8*x^-log2 x = 6. What is x?

We have to solve for x given that x^log2 x + 8*x^-log2 x = 6 x^log(2) x + 8*x^(- log(2) x) = 6 => x^log(2) x + 8/x^(log(2) x) = 6 Let x^log(2) x = y => y + 8/y = 6 => y^2 - 6x + 8 = 0...

algebra1

### Find the inverse of f(x)=5tan(3x+4)

The first step in finding an inverse function is to write the function in “y equals” notation, as it will allow us simply to switch the variables and solve for us once again as follows: Let...

algebra1

### What kind of equation is z^4-3z^2+2=0 ?

The equation you have given is z^4 - 3z^2 + 2 = 0. Here you can replace z^2 = x to get a quadratic equation in x which can be solved. z^4 - 3z^2 + 2 = 0 => x^2 - 3x + 2 = 0 => x^2 - 2x - x +...

algebra1

### What is x if x^3-1728=0?

We are given that x^3 - 1728 = 0 and we have to find x. x^3 - 1728 = 0 => x^3 = 1728 => x^3 = 12^3 As the exponent is the same, we can equate the bases This gives x = 12.

algebra1

### Simplified product Simplify (8-6i)(-4-4i).

To simplify (8-6i)(-4-4i) open the brackets and multiply the terms (8-6i)(-4-4i) => 8*-4 - 8*4i - 6i*(-4) - 6i*(-4i) => -32 - 32i + 24i + 24i^2 => -32 - 8i - 24 => -56 - 8i The required...

algebra1

### How can I use substitution in the problem 2x-y=5, 3x-2y=9 ? How can I use substitution in the problem 2x-y=5, 3x-2y=9 ?

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

### composition of functions Find f(f(4)) if f(x)=14x+13.

It is given that f(x)=14x+13. f(f(4)) given that f(4) = 14*4 + 13 => f( 14*4 + 13) => f(69) => 14*69 + 13 => 979 The value of f(f(4)) = 979

algebra1

### I don't know to find parabola mx^2+2mx+x+m-1. all i know is the extreme of parabola is on the line 9x-3y-3.

The extreme of the parabola is determined by finding the first derivative and equating it to 0. We then solve for x. y = mx^2+2mx+x+m-1 y' = 2mx + 2m + 1 = 0 => 2mx = -2m - 1 => x = (-2m - 1)...

algebra1

### If x=1 is the root of 5x^3-4x^2+7x-8=0, what are the other roots of equation? (use the remainder theorem)

Using the remainder theorem, as x = 1 is a root of the expression 5x^3-4x^2+7x-8=0, the expression is divisible by ( x - 1). (ax^2 + bx + c)(x - 1) = 5x^3-4x^2+7x-8 => ax^3 + bx^2 + cx - ax^2 -...

algebra1

### What is the sum of the 12 terms of AP if a1+a5+a8+a12=24 ?

For an AP the nth terms can be written as a + (n-1)*d, where a is the first term and d is the common difference between consecutive terms. The sum of the first n terms is (t1 + tn)*(n/2) In the...

algebra1

### Determine the complex number z if (3z-2z')/6=-5? z' is the conjugate of z.

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

### If x and y are 2 real numbers such that 5x+4y=9 and 3x+2y=5, then what is 4x+3y?

We have 5x + 4y = 9 and 3x + 2y = 5. We need to find 4x + 3y. Here we don't need to find x and y. The required result can be obtained by adding 5x + 4y = 9 and 3x + 2y = 5 => 5x + 4y + 3x + 2y =...

algebra1

### System. Solve the system -2x-y=-9 5x-2y=18

We have to solve -2x - y = -9 ...(1) 5x - 2y = 18 ...(2) 2*(1) - (2) => -4x - 2y - 5x + 2y = -18 - 18 => -9x = -36 => x = 4 substitute in (1) -2x - y = -9 => y = -2*4 + 9 => y = 1...

algebra1

### Determine the numbers a,b if the law of composition x*y=xy+2ax+by is commutative.

We have to determine a and b given that for x*y = xy + 2ax + by, the * operator is commutative. x*y = y*x => xy + 2ax + by = yx + 2ay + bx => 2ax + by = 2ay + bx equating the coefficients of...

algebra1

### Solve the simultaneous equations y^2=x^2-9 and y= x-1 .

We have to solve the equations: y^2 = x^2 - 9 ...(1) y = x - 1 ...(2) From (2) we get y = x - 1 => y^2 = (x - 1)^2 substitute in (1) (x - 1)^2 = x^2 - 9 => x^2 + 1 - 2x = x^2 - 9 => -2x =...

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