Hard Algebra Worksheets 100 Problems
Algebra worksheets can be a valuable tool for students, providing them with ample practice to strengthen their understanding of this challenging subject. With a diverse range of exercises, these worksheets focus on various algebraic concepts, equipping students with the necessary skills to tackle 100 problems with confidence and proficiency.
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What is the value of x if 3x + 5 = 17?
The value of x is 4. This can be found by isolating x in the equation 3x + 5 = 17. Subtracting 5 from both sides gives 3x = 12, and then dividing by 3 gives x = 4.
Solve the equation 2(4x + 3) = 10x - 6.
To solve the equation 2(4x + 3) = 10x - 6, first distribute the 2 on the left side: 8x + 6 = 10x - 6. Then, move all the terms with x to one side by subtracting 8x from both sides: 6 = 2x - 6. Next, add 6 to both sides: 12 = 2x. Finally, divide by 2 to solve for x: x = 6.
Simplify the expression 5(2x - 3) - 4(x + 2).
To simplify the expression 5(2x - 3) - 4(x + 2), first distribute the 5 and 4 to the terms inside the parentheses: 10x - 15 - 4x - 8. Then, combine like terms to simplify further: 10x - 4x - 15 - 8 = 6x - 23. Thus, the simplified expression is 6x - 23.
Find the roots of the quadratic equation x^2 + 6x - 7 = 0.
To find the roots of the quadratic equation x^2 + 6x - 7 = 0, we can use the quadratic formula: x = (-b ± ?(b^2 - 4ac)) / 2a. Here, a = 1, b = 6, and c = -7. By substituting these values into the formula, we get x = (-6 ± ?(6^2 - 4*1*(-7))) / 2*1. Simplifying this further, x = (-6 ± ?(36 + 28)) / 2, x = (-6 ± ?64) / 2, x = (-6 ± 8) / 2. Therefore, the roots are x = (-6 + 8) / 2 and x = (-6 - 8) / 2, which simplifies to x = 1 and x = -7. The roots of the quadratic equation x^2 + 6x - 7 = 0 are x = 1 and x = -7.
Simplify the expression (2x^3 - 3x^2 + x) + (4x^2 + 5x - 2).
The simplified expression is 2x^3 + x + x^2 + 5x - 2.
Solve the equation 3(2y + 1) - 4(y - 3) = 8y - 5.
To solve the equation 3(2y + 1) - 4(y - 3) = 8y - 5, we need to first distribute the terms inside the parentheses. This gives us 6y + 3 - 4y + 12 = 8y - 5. Simplifying further, we combine like terms to get 2y + 15 = 8y - 5. Next, we isolate the variable terms on one side, so subtract 2y from both sides to get 15 = 6y - 5. Adding 5 to both sides, we have 20 = 6y. Finally, divide by 6 to solve for y, giving us y = 20/6 or y = 10/3.
Factor the expression 9x^2 - 24x + 16.
To factor the expression 9x^2 - 24x + 16, we first look for two numbers that multiply to 9*16=144 and add up to -24. The numbers that fit this criteria are -12 and -12. Therefore, the factored form of 9x^2 - 24x + 16 is (3x - 4)^2.
Solve the system of equations: 2x + 3y = 4 and 4x - 2y = 10.
To solve the system of equations 2x + 3y = 4 and 4x - 2y = 10, we can first multiply the first equation by 2 to get 4x + 6y = 8. Then, we can add the two equations together to eliminate the y variable: 4x + 6y + 4x - 2y = 8 + 10, which simplifies to 8x + 4 = 18. Solve for x by isolating x: 8x = 14, x = 14/8 = 7/4 or 1.75. Substitute x back into either equation, like 2x + 3y = 4, to solve for y: 2(1.75) + 3y = 4, 3.5 + 3y = 4, 3y = 0.5, y = 0.5/3 = 1/6 or approximately 0.167. Therefore, the solution to the system of equations is x = 1.75 and y = 0.167.
Simplify the expression (a^2 - b^2)/(a^2 + 2ab + b^2).
The expression can be simplified to (a - b)/(a + b).
Find the value of y if (3y + 2)^2 = 49.
To find the value of y, we can first expand the expression (3y + 2)^2 as (3y + 2)(3y + 2) = 3y(3y) + 2(3y) + 3y(2) + 2(2) = 9y^2 + 6y + 6y + 4 = 9y^2 + 12y + 4. Now, we set this equal to 49: 9y^2 + 12y + 4 = 49. Simplifying further, we get 9y^2 + 12y - 45 = 0. Factoring this quadratic equation, we have (3y + 15)(3y - 3) = 0, which gives us y = -5 or y = 1. Thus, the values of y are -5 and 1.
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