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Quadratic equations

Solve equations like x² − 5x + 6 = 0 by factoring, find the sum of the solutions, and use the special products of a and b.

A quadratic equation is an equation with an x² in it, like x² − 5x + 6 = 0. The name comes from the Latin word for a square. People have solved equations like this for almost 4,000 years: clay tablets from ancient Babylon show problems about fields whose side and area were mixed together, and their answers. Today quadratics describe the path of a thrown ball and the shape of some bridge arches. The good news is that the ones in everyday puzzles can be solved with one clear idea and a little detective work.

The key idea: a product that is zero

If two numbers multiply to give 0, at least one of them must be 0. There is no other way to get 0. So if (x − 2)(x − 3) = 0, then x − 2 = 0 or x − 3 = 0. That gives x = 2 or x = 3.

Remember

If AB = 0, then A = 0 or B = 0.

This is why we try to write a quadratic as two brackets multiplied together. That is called factoring. Once it is factored, each bracket gives one solution. A quadratic usually has two solutions, and the solutions are also called roots.

How the brackets make the equation

To learn how to factor, first go the other way. Multiply out (x + 2)(x + 3) term by term. Each term in the first bracket multiplies each term in the second bracket.

Example 1

Multiply out (x + 2)(x + 3).

  1. x times x gives x².
  2. x times 3 gives 3x.
  3. 2 times x gives 2x.
  4. 2 times 3 gives 6.
  5. Together: x² + 3x + 2x + 6 = x² + 5x + 6.

Answer: x² + 5x + 6

Look at where the numbers went. The 2 and the 3 add to make the middle number 5. They multiply to make the last number 6. This is true for every pair of brackets like these, and it is the whole secret of factoring.

Remember

To factor x² + bx + c, find two numbers that multiply to c and add to b.

Solving by factoring

  1. Make sure the equation has 0 on one side.
  2. List pairs of numbers that multiply to the last number.
  3. Pick the pair that adds to the middle number. Watch the signs.
  4. Write the two brackets.
  5. Set each bracket equal to 0 and solve.
  6. Check each answer in the first equation.

Example 2

Solve x² − 7x + 10 = 0.

  1. We need two numbers that multiply to 10 and add to −7.
  2. Pairs that multiply to 10: 1 and 10, 2 and 5, −1 and −10, −2 and −5.
  3. Only −2 and −5 add to −7.
  4. So (x − 2)(x − 5) = 0.
  5. x − 2 = 0 gives x = 2. x − 5 = 0 gives x = 5.
  6. Check x = 2: 4 − 14 + 10 = 0. Check x = 5: 25 − 35 + 10 = 0.

Answer: x = 2 or x = 5

Example 3

Solve x² − 2x − 8 = 0.

  1. We need two numbers that multiply to −8 and add to −2.
  2. The product is negative, so one number is positive and one is negative.
  3. 2 and −4 work: 2(−4) = −8 and 2 + (−4) = −2.
  4. So (x + 2)(x − 4) = 0.
  5. x + 2 = 0 gives x = −2. x − 4 = 0 gives x = 4.
  6. Check x = −2: 4 + 4 − 8 = 0. Check x = 4: 16 − 8 − 8 = 0.

Answer: x = −2 or x = 4

Example 4

Solve x² + 8x + 15 = 0.

  1. Two numbers that multiply to 15 and add to 8: they are 3 and 5.
  2. So (x + 3)(x + 5) = 0.
  3. x + 3 = 0 gives x = −3. x + 5 = 0 gives x = −5.
  4. Check x = −3: 9 − 24 + 15 = 0.

Answer: x = −3 or x = −5

The sum and the product of the solutions

Some questions do not ask for the solutions themselves. They ask for their sum (the two added) or their product (the two multiplied). The plain way is to solve the equation first, then add or multiply.

Example 5

What is the sum of the two solutions of x² − 10x + 21 = 0?

  1. Two numbers that multiply to 21 and add to −10: −3 and −7.
  2. So (x − 3)(x − 7) = 0, and the solutions are 3 and 7.
  3. Their sum: 3 + 7 = 10.

Answer: 10

Example 6

What is the product of the two solutions of x² + 3x − 10 = 0?

  1. Two numbers that multiply to −10 and add to 3: 5 and −2.
  2. So (x + 5)(x − 2) = 0, and the solutions are −5 and 2.
  3. Their product: (−5)(2) = −10.

Answer: −10

Finding a squared plus b squared

Here is a different kind of question. You are told a + b = 7 and ab = 10, and asked for a² + b². You do not need to find a and b. Instead, multiply out (a + b)², which means (a + b)(a + b), term by term.

  1. a times a gives a².
  2. a times b gives ab.
  3. b times a gives ab again.
  4. b times b gives b².
  5. Together: (a + b)² = a² + 2ab + b².

Remember

(a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²

Example 7

a + b = 7 and ab = 10. What is a² + b²?

  1. (a + b)² = a² + 2ab + b².
  2. Put in what we know: 7² = a² + b² + 2(10).
  3. 49 = a² + b² + 20.
  4. Take 20 away from both sides: a² + b² = 29.
  5. Check with a = 2 and b = 5: they add to 7, multiply to 10, and 4 + 25 = 29.

Answer: 29

Example 8

x − y = 5 and xy = 3. What is x² + y²?

  1. (x − y)² = x² − 2xy + y².
  2. Put in what we know: 5² = x² + y² − 2(3).
  3. 25 = x² + y² − 6.
  4. Add 6 to both sides: x² + y² = 31.

Answer: 31

The difference of two squares

Now multiply out (a + b)(a − b) term by term: a times a gives a², a times −b gives −ab, b times a gives ab, and b times −b gives −b². The −ab and the ab cancel. Only a² − b² is left.

Remember

a² − b² = (a + b)(a − b)

A square minus a square is the sum times the difference.

Example 9

x + y = 10 and x − y = 4. What is x² − y²?

  1. x² − y² = (x + y)(x − y).
  2. Put in the two numbers: (10)(4).
  3. 10 · 4 = 40.

Answer: 40

Example 10

Work out 41² − 39² without a calculator.

  1. 41² − 39² = (41 + 39)(41 − 39).
  2. 41 + 39 = 80 and 41 − 39 = 2.
  3. 80 · 2 = 160.
  4. Check: 1681 − 1521 = 160.

Answer: 160

Try it yourself

  1. 1.Solve x² − 6x + 8 = 0.

    Show the answer

    x = 2 or x = 4

    −2 and −4 multiply to 8 and add to −6, so (x − 2)(x − 4) = 0.

  2. 2.Solve x² + x − 12 = 0.

    Show the answer

    x = −4 or x = 3

    4 and −3 multiply to −12 and add to 1, so (x + 4)(x − 3) = 0.

  3. 3.What is the sum of the two solutions of x² − 11x + 24 = 0?

    Show the answer

    11

    It factors as (x − 3)(x − 8), so the solutions are 3 and 8, and 3 + 8 = 11.

  4. 4.a + b = 6 and ab = 8. What is a² + b²?

    Show the answer

    20

    (a + b)² = a² + b² + 2ab, so 36 = a² + b² + 16, and a² + b² = 20.

  5. 5.x − y = 3 and xy = 10. What is x² + y²?

    Show the answer

    29

    (x − y)² = x² + y² − 2xy, so 9 = x² + y² − 20, and x² + y² = 29.

  6. 6.Work out 31² − 29².

    Show the answer

    120

    (31 + 29)(31 − 29) = 60 · 2 = 120.

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