Hard algebra practice with explanations

Algebra Problems with Step-by-Step Solutions

Practice challenging algebra problems from intermediate to super hard level. Each problem includes hints, answers, and clear explanations so you can improve problem-solving skills, not just memorize formulas.

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Algebraic expressions Hard

If a + b = 7 and ab = 10, find a² + b².

  1. Use the identity: (a + b)² = a² + 2ab + b².
  2. Substitute the known values: 7² = a² + 2 · 10 + b².
  3. So 49 = a² + b² + 20.
  4. Subtract 20 from both sides.
Correct answer: a² + b² = 29
Systems of equations Advanced

Solve the system x + y = 6 and x² + y² = 20.

  1. Square the first equation: (x + y)² = 36.
  2. This gives x² + 2xy + y² = 36.
  3. Substitute x² + y² = 20, so 20 + 2xy = 36.
  4. Then xy = 8. The numbers x and y have sum 6 and product 8.
  5. So they are roots of t² − 6t + 8 = 0.
Correct answer: (x, y) = (2, 4) or (4, 2)
Quadratics College

For what values of k does x² − kx + 9 = 0 have one repeated root?

  1. A quadratic has one repeated root when its discriminant equals zero.
  2. For ax² + bx + c, the discriminant is b² − 4ac.
  3. Here, a = 1, b = −k, and c = 9.
  4. So the discriminant is k² − 36.
  5. Set k² − 36 = 0 and solve.
Correct answer: k = 6 or k = −6
Interactive test preview

Try one algebra question

This is a small demo of the future test format: 4 answer choices, instant checking, and a short explanation.

If 2x − 3 = 11, what is the value of x?

Explanation: Add 3 to both sides: 2x = 14. Divide by 2, so x = 7.
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