Elementary Algebra Flashcards

√ 4 × √ 5 =

Answer: 2 √ 5

To solve this, first evaluate the square root of 4, which is 2. Then, multiply this result by the square root of 5. Since the square root of 5 cannot be simplified further into an integer, the expression becomes 2 times the square root of 5.

4(| − 3 − 2|) − 5 =

Answer: 15

First, calculate the value inside the absolute value bars: -3 - 2 equals -5. The absolute value of -5 is 5. Next, multiply this result by 4, which gives 20. Finally, subtract 5 from 20 to get the final answer of 15.

−2⁄3 + 1⁄6 ⋅ (−2) = ____

Answer: -1

Following the order of operations (PEMDAS/BODMAS), first perform the multiplication: (1/6) * (-2) = -2/6, which simplifies to -1/3. Now the expression is -2/3 + (-1/3). Adding these fractions with a common denominator gives -3/3, which simplifies to -1.

Which of the following sequence of numbers lists the numbers from the least to the greatest?

Answer: −5⁄6 2⁄3∣ 1⁄2 = 4⁄8 7⁄2

To order these numbers, first convert them to decimals or fractions with a common denominator. -|-2/3| simplifies to -2/3 (approximately -0.667). -5/6 is approximately -0.833. 1/2 is 0.5, and 4/8 is also 0.5. 7/2 is 3.5. Arranging these values from least to greatest gives -0.833 < -0.667 < 0.5 = 0.5 < 3.5, which corresponds to -5/6 < -|-2/3| < 1/2 = 4/8 < 7/2.

4(−7+5) − (−2)(6−11) = ____

Answer: -18

Following the order of operations, first solve the expressions within the parentheses. (-7 + 5) equals -2, and (6 - 11) equals -5. Now substitute these values back into the equation: 4(-2) - (-2)(-5). Perform the multiplications: 4(-2) = -8, and (-2)(-5) = 10. Finally, subtract the results: -8 - 10 = -18.

x(x2−3)−2x2+5 =

Answer: x3 − 2x2− 3x + 5

To simplify the expression, first distribute x into the parentheses: x * x^2 = x^3 and x * -3 = -3x. This transforms the expression into x^3 - 3x - 2x^2 + 5. Finally, rearrange the terms in descending order of their exponents to get the standard polynomial form: x^3 - 2x^2 - 3x + 5.

(5x+2y)2 =

Answer: 25x2 + 20xy+4y2

To expand (5x+2y)^2, use the formula (a+b)^2 = a^2 + 2ab + b^2. Here, a = 5x and b = 2y. So, a^2 = (5x)^2 = 25x^2, b^2 = (2y)^2 = 4y^2, and 2ab = 2(5x)(2y) = 20xy. Combining these terms gives the expanded form: 25x^2 + 20xy + 4y^2.

(x2+ 8x + 15)⁄(x + 3) =

Answer: (x+5)

To simplify this rational expression, factor the quadratic in the numerator. We need two numbers that multiply to 15 and add to 8, which are 3 and 5. So, x^2 + 8x + 15 factors into (x+3)(x+5). Now, the expression becomes [(x+3)(x+5)] / (x+3). Since (x+3) is in both the numerator and denominator, they cancel out, leaving (x+5).

xy−2y2+5y=4

Answer: x=2y+ 4⁄y −5

To solve for x, first isolate the term containing x. Add 2y^2 and subtract 5y from both sides of the equation, resulting in xy = 4 + 2y^2 - 5y. Then, divide the entire right side by y to solve for x. This yields x = (4 + 2y^2 - 5y) / y, which can be rewritten as x = 4/y + 2y - 5, or x = 2y + 4/y - 5.

Evaluate the following expression for x = 6 and y = -1: 3y2−4x+2xy

Answer: -33

Substitute the given values x=6 and y=-1 into the expression. This gives 3(-1)^2 - 4(6) + 2(6)(-1). Calculate each term: 3(1) = 3, -4(6) = -24, and 2(6)(-1) = -12. Finally, add these results: 3 - 24 - 12 = -21 - 12 = -33.

Solve the equation for x, given y = 3 9x−12+y2 = 5

Answer: x = 8⁄9

First, substitute the value of y=3 into the equation: 9x - 12 + (3)^2 = 5. This simplifies to 9x - 12 + 9 = 5. Combine the constant terms: 9x - 3 = 5. Add 3 to both sides to isolate the x term: 9x = 8. Finally, divide by 9 to solve for x: x = 8/9.

What are the roots of the polynomial (x+1)(x2−x−6) = 0?

Answer: −1,−2 and 3

To find the roots, set each factor equal to zero. From (x+1)=0, we get x=-1. For the quadratic factor, x^2 - x - 6 = 0, factor it into (x-3)(x+2) = 0. Setting each of these factors to zero gives x-3=0 (so x=3) and x+2=0 (so x=-2). Therefore, the roots are -1, -2, and 3.

(-3)(2+6)⁄4 - 2 =

Answer: -8

Following the order of operations (PEMDAS/BODMAS), first solve the expression inside the parentheses: 2 + 6 = 8. Next, perform the multiplication in the numerator: (-3)(8) = -24. Then, divide by 4: -24 / 4 = -6. Finally, subtract 2 from the result: -6 - 2 = -8.

Which of the following is equal to (x+3)(x2+3x−5)?

Answer: x3+6x2+4x−15

To expand the expression, use the distributive property (multiply each term in the first parenthesis by each term in the second). Multiply x by (x^2 + 3x - 5) to get x^3 + 3x^2 - 5x. Then, multiply 3 by (x^2 + 3x - 5) to get 3x^2 + 9x - 15. Combine these results: x^3 + 3x^2 - 5x + 3x^2 + 9x - 15. Finally, combine like terms: x^3 + (3x^2 + 3x^2) + (-5x + 9x) - 15, which simplifies to x^3 + 6x^2 + 4x - 15.

x = 3 and x = -3 are both solutions to which of the following equations?

Answer: x2-9 = 0

To find the correct equation, substitute x=3 and x=-3 into each option. For option D, x² - 9 = 0, substituting x=3 gives 3² - 9 = 9 - 9 = 0, which is true. Substituting x=-3 gives (-3)² - 9 = 9 - 9 = 0, which is also true. This equation represents the difference of squares, (x-3)(x+3)=0, whose roots are indeed 3 and -3.

For x > 0, 3y⁄x - y⁄2x + 2y⁄4x =

Answer: 3y⁄x

To combine these fractions, first simplify the third term: 2y/4x simplifies to y/2x. Now the expression is 3y/x - y/2x + y/2x. The terms -y/2x and +y/2x cancel each other out. Therefore, the simplified expression is simply 3y/x.

Which of the following correctly lists the numbers in order from greatest to least?ub>2

Answer: 5⁄2 > 3⁄4 > − 1⁄4 > −1⁄2

To order the numbers, it's helpful to convert them to decimals: 5/2 = 2.5, 3/4 = 0.75, -1/4 = -0.25, and -1/2 = -0.5. Arranging these from greatest to least gives 2.5 > 0.75 > -0.25 > -0.5. This corresponds to 5/2 > 3/4 > -1/4 > -1/2.

Which of the following lists the fractions in order from greatest to least when x = -1?

Answer: −3⁄2x > 2⁄3 > 1⁄2x > 2x

First, substitute x = -1 into each expression: 2x becomes 2(-1) = -2; 1/2x becomes 1/2(-1) = -1/2; -3/2x becomes -3/2(-1) = 3/2; and 2/3 remains 2/3. Now, convert these values to decimals for easier comparison: -2, -0.5, 1.5, and 0.66... Ordering these from greatest to least gives 1.5 > 0.66... > -0.5 > -2, which corresponds to -3/2x > 2/3 > 1/2x > 2x.

3 - 7x⁄2 < 10

Answer: x > −2

To solve the inequality, first subtract 3 from both sides: -7x/2 -2.

Solve the following system of equations for (x,y), 3x+4y = 25, x−2y = 5

Answer: (7, 1)

Using the elimination method, multiply the second equation (x - 2y = 5) by 2 to get 2x - 4y = 10. Now add this new equation to the first equation (3x + 4y = 25): (3x + 4y) + (2x - 4y) = 25 + 10, which simplifies to 5x = 35. Dividing by 5 gives x = 7. Substitute x = 7 into the second original equation: 7 - 2y = 5. Subtract 7 from both sides: -2y = -2. Dividing by -2 gives y = 1. Thus, the solution is (7, 1).