Skip to content

Alerts Center

New quiz added: Multiplication Tables — try it now

Fractions lesson updated with new practice worksheets

This month's resource pack is now live

View All Alerts

Messages

Tip: Create a free account to save your quiz progress

New here? Check out our Getting Started guide

New Sudoku puzzles added this week

View All Messages

Algebra Tricks

Solving Quadratic Equations – Sum and Product of Roots

Solve a quadratic in your head: no formula, only a sum, a product and one square root.

This page explains the rule, shows why it works, and walks through worked examples. Then you can try it yourself and check your understanding with practice questions. Use the previous and next links at the bottom to move through the tricks in order.

The rule

The Rule: In x2 + bx + c = 0 the two roots add up to −b and multiply to c. Write them as m + a and m − a, where m is half the sum. Then m2 − a2 = c, so a2 = m2 − c, and the roots are m + a and m − a.

Why it works

Why It Works: A quadratic with roots p and q can always be written as (x − p)(x − q). Multiplying that out gives x2 − (p + q)x + pq, so the number in front of x is minus the sum of the roots, and the last number is their product. That is why the sum and product are visible straight from the equation. Next, the two roots sit the same distance, a, either side of their midpoint m (half the sum). Multiplying (m + a)(m − a) makes the middle terms cancel, leaving m2 − a2. That leaves only one unknown, a, and it drops out with one subtraction and one square root, with no formula to memorise.

Formal Proof (for advanced learners)

Let the roots of x2 + bx + c = 0 be p and q. Then x2 + bx + c = (x − p)(x − q) = x2 − (p + q)x + pq, so p + q = −b and pq = c. Let m = (p + q)/2 = −b/2 and a = (p − q)/2, so p = m + a and q = m − a. Then c = pq = (m + a)(m − a) = m2 − a2, so a = √(m2 − c). Written out, x = −b/2 ± √(b2/4 − c), which is the quadratic formula for an equation whose x2 coefficient is 1. The trick is the formula in disguise.

Examples

  • x2 − 14x + 45 = 0 → sum 14, product 45 → 7 ± a → 49 − a2 = 45 → a = 2 → x = 5 or x = 9.
  • x2 − 10x + 21 = 0 → sum 10, product 21 → 5 ± a → 25 − a2 = 21 → a = 2 → x = 3 or x = 7.
  • x2 + 8x + 15 = 0 → sum −8, product 15 → −4 ± a → 16 − a2 = 15 → a = 1 → x = −5 or x = −3.

Step by step

Example A: solve x2 − 14x + 45 = 0

  1. Compare with x2 + bx + c = 0. The roots add up to −b, so they add up to 14, and they multiply to c, which is 45.
  2. Half the sum is 14 ÷ 2 = 7. The two roots sit the same distance, a, either side of 7, so write them as 7 + a and 7 − a.
  3. Multiply them: (7 + a)(7 − a) = 49 − a2. The middle terms cancel, and the result must equal the product, 45.
  4. So 49 − a2 = 45, which means a2 = 49 − 45 = 4. The positive number whose square is 4 is a = 2.
  5. Put a back in: 7 + 2 = 9 and 7 − 2 = 5. So the roots are 9 and 5.
  6. Check: 9 + 5 = 14, which matches the sum, and 9 × 5 = 45, which matches the product.

Example B: solve x2 + 10x + 24 = 0

  1. Compare with x2 + bx + c = 0. The roots add up to −b, so they add up to −10, and they multiply to c, which is 24.
  2. Half the sum is −10 ÷ 2 = −5. The two roots sit the same distance, a, either side of −5, so write them as −5 + a and −5 − a.
  3. Multiply them: (−5 + a)(−5 − a) = 25 − a2. The middle terms cancel, and the result must equal the product, 24.
  4. So 25 − a2 = 24, which means a2 = 25 − 24 = 1. The positive number whose square is 1 is a = 1.
  5. Put a back in: −5 + 1 = −4 and −5 − 1 = −6. So the roots are −4 and −6.
  6. Check: −4 + −6 = −10, which matches the sum, and −4 × −6 = 24, which matches the product.

Try it yourself

Now it is your turn. A new random question appears each time. Type your answer in the box and press Check Answer. If you are right, we show how the trick got you there, step by step. If you are not, you will see the full working two ways, the shortcut and the usual method, so you can spot exactly where your answer differed. Press New Question as often as you like: a few quick rounds is the best way to make the trick feel automatic.

Try It Yourself

Practice questions

Have a go on paper first, then tap Show answer to check your method.

  1. Solve x2 − 10x + 21 = 0.

    Show answer

    x = 3 or x = 7. Compare with x2 + bx + c = 0. The roots add up to −b, so they add up to 10, and they multiply to c, which is 21. Half the sum is 10 ÷ 2 = 5. The two roots sit the same distance, a, either side of 5, so write them as 5 + a and 5 − a. Multiply them: (5 + a)(5 − a) = 25 − a2. The middle terms cancel, and the result must equal the product, 21. So 25 − a2 = 21, which means a2 = 25 − 21 = 4. The positive number whose square is 4 is a = 2. Put a back in: 5 + 2 = 7 and 5 − 2 = 3. So the roots are 7 and 3. Check: 7 + 3 = 10, which matches the sum, and 7 × 3 = 21, which matches the product.

  2. Solve x2 − 16x + 63 = 0.

    Show answer

    x = 7 or x = 9. Compare with x2 + bx + c = 0. The roots add up to −b, so they add up to 16, and they multiply to c, which is 63. Half the sum is 16 ÷ 2 = 8. The two roots sit the same distance, a, either side of 8, so write them as 8 + a and 8 − a. Multiply them: (8 + a)(8 − a) = 64 − a2. The middle terms cancel, and the result must equal the product, 63. So 64 − a2 = 63, which means a2 = 64 − 63 = 1. The positive number whose square is 1 is a = 1. Put a back in: 8 + 1 = 9 and 8 − 1 = 7. So the roots are 9 and 7. Check: 9 + 7 = 16, which matches the sum, and 9 × 7 = 63, which matches the product.

  3. Solve x2 − 20x + 96 = 0.

    Show answer

    x = 8 or x = 12. Compare with x2 + bx + c = 0. The roots add up to −b, so they add up to 20, and they multiply to c, which is 96. Half the sum is 20 ÷ 2 = 10. The two roots sit the same distance, a, either side of 10, so write them as 10 + a and 10 − a. Multiply them: (10 + a)(10 − a) = 100 − a2. The middle terms cancel, and the result must equal the product, 96. So 100 − a2 = 96, which means a2 = 100 − 96 = 4. The positive number whose square is 4 is a = 2. Put a back in: 10 + 2 = 12 and 10 − 2 = 8. So the roots are 12 and 8. Check: 12 + 8 = 20, which matches the sum, and 12 × 8 = 96, which matches the product.

  4. Solve x2 + 8x + 15 = 0.

    Show answer

    x = −5 or x = −3. Compare with x2 + bx + c = 0. The roots add up to −b, so they add up to −8, and they multiply to c, which is 15. Half the sum is −8 ÷ 2 = −4. The two roots sit the same distance, a, either side of −4, so write them as −4 + a and −4 − a. Multiply them: (−4 + a)(−4 − a) = 16 − a2. The middle terms cancel, and the result must equal the product, 15. So 16 − a2 = 15, which means a2 = 16 − 15 = 1. The positive number whose square is 1 is a = 1. Put a back in: −4 + 1 = −3 and −4 − 1 = −5. So the roots are −3 and −5. Check: −3 + −5 = −8, which matches the sum, and −3 × −5 = 15, which matches the product.

  5. The perimeter of a rectangle is 36 cm and its area is 77 cm2. Find its length and width.

    Show answer

    Length 11 cm and width 7 cm. Half the perimeter is length + width = 18 cm, and length × width = 77. So the two sides are the roots of x2 − 18x + 77 = 0. Compare with x2 + bx + c = 0. The roots add up to −b, so they add up to 18, and they multiply to c, which is 77. Half the sum is 18 ÷ 2 = 9. The two roots sit the same distance, a, either side of 9, so write them as 9 + a and 9 − a. Multiply them: (9 + a)(9 − a) = 81 − a2. The middle terms cancel, and the result must equal the product, 77. So 81 − a2 = 77, which means a2 = 81 − 77 = 4. The positive number whose square is 4 is a = 2. Put a back in: 9 + 2 = 11 and 9 − 2 = 7. So the roots are 11 and 7. Check: 11 + 7 = 18, which matches the sum, and 11 × 7 = 77, which matches the product.

Tips and watch-outs

  • The sum of the roots is the number in front of x with its sign flipped. In x2 + 14x + 45 = 0 the roots add up to −14, so both are negative: −5 and −9.
  • This works when the roots are whole numbers and the x2 term has coefficient 1. If the equation starts with 2x2 or 3x2, divide every term by that number first.
  • If m2 − c is not a perfect square, the roots are not whole numbers. Use the quadratic formula instead (it is the same method: a = √(m2 − c)).
  • An odd sum works too, only m ends in .5: for x2 − 7x + 12 = 0, m = 3.5, a = 0.5, so the roots are 4 and 3.