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➕ Algebra 1 · Quadratics

Solving Quadratics by Factoring: notes and common mistakes

Study notes

Key ideas

  • A quadratic equation can be written ax2 + bx + c = 0 with a ≠ 0. It has at most two real solutions.
  • Zero-product property: if A × B = 0, then A = 0 or B = 0 (or both). It works only when the product equals 0, so get 0 on one side first.
  • Steps: zero on one side, factor completely, set each factor to 0, solve each one, and check in the original equation.
  • A repeated factor gives a double root (one solution). A common factor of x gives the solution x = 0. A constant factor gives no solution.
  • Never divide both sides by an expression that could be 0, such as x. It loses solutions.
  • If a trinomial does not factor with integers, the equation may still have solutions. They need later methods.
  • The solutions of ax2 + bx + c = 0 are the x-intercepts of the graph of y = ax2 + bx + c.

Steps

  1. Write the equation as ax2 + bx + c = 0 (every term on one side, 0 on the other).
  2. Factor completely: GCF first, then trinomial, difference of squares, and so on.
  3. Set each factor that contains x equal to 0.
  4. Solve each equation.
  5. Check each solution in the original equation. In a word problem, also check it against the situation.

Formulas

  • Standard form of the equation: ax2 + bx + c = 0, with a ≠ 0
  • Zero-product property: If A × B = 0, then A = 0 or B = 0 (or both)
  • Difference of squares: a2 − b2 = (a + b)(a − b)
  • Perfect square trinomials: a2 + 2ab + b2 = (a + b)2 and a2 − 2ab + b2 = (a − b)2 (each gives a double root)
  • Solutions and the graph: Solutions of ax2 + bx + c = 0 = x-intercepts of y = ax2 + bx + c

Watch out

  • Get 0 on one side before you factor and split. (x − 5)(x − 2) = 10 does not mean x − 5 = 10 or x − 2 = 10.
  • Do not divide both sides by x. Move everything to one side and factor out x, so you keep x = 0.
  • Flip the sign: for a factor like x + 9, the solution is x = −9.
  • Factor completely before you split. Take out the GCF first.
  • A double root is one solution, not two.
  • For x2 = 49 or 25x2 − 9 = 0, there are two solutions, one positive and one negative.
  • Not factoring with integers does not mean no solution.
  • In a word problem, reject a solution that does not fit the story (such as a negative length) and say why.

Why it works

  • Why does the zero-product property work? If A × B = 0 and A ≠ 0, divide both sides by A to get B = 0. So either A = 0 or, if not, B = 0.
  • Why must the other side be 0? Only 0 has this power. Many pairs of numbers multiply to 12, so a product of 12 does not force either factor to be any particular number. A product of 0 forces one of them to be 0.
  • Why are the solutions the x-intercepts? An x-intercept is a point of the graph with y = 0. Putting y = 0 into y = ax2 + bx + c gives ax2 + bx + c = 0, the equation we solved.
  • Why does a perfect square give a double root? In (x + 4)2 = 0 the factor x + 4 appears twice, so setting it to 0 gives only one value of x. The graph stays on or above the x-axis, so it only touches it.

Common mistakes

✗ MistakeIf (x − 5)(x − 3) = 8, then x − 5 = 8 or x − 3 = 8.
✓ InsteadThe zero-product property works only when the product is 0. Move the 8 first: x2 − 8x + 7 = 0, so (x − 1)(x − 7) = 0 and x = 1 or x = 7.
✗ MistakeTo solve x2 = 5x, divide both sides by x to get x = 5.
✓ InsteadDividing by x is only allowed when x ≠ 0, and it loses the solution x = 0. Write x2 − 5x = 0 and factor: x(x − 5) = 0, so x = 0 or x = 5.
✗ MistakeThe solutions of (x + 3)(x − 7) = 0 are x = 3 and x = −7.
✓ InsteadSet each factor to 0: x + 3 = 0 gives x = −3, and x − 7 = 0 gives x = 7. The solution is the number that makes the factor 0, so the sign flips.
✗ MistakeIf a quadratic equation cannot be factored with integers, it has no solution.
✓ InsteadIt may still have solutions, which are not whole numbers. Later lessons (completing the square and the quadratic formula) find them.
✗ MistakeIn an area problem, keep both answers, even a negative one, because both satisfy the equation.
✓ InsteadBoth solve the equation, but only the one that fits the story answers the question. A negative length is impossible, so reject it and say why.
✗ Mistake(x − 6)2 = 0 has two solutions, x = 6 and x = −6.
✓ InsteadBoth factors are x − 6, so there is only one solution, x = 6, a double root. The value x = −6 gives (−12)2 = 144, not 0.

Teach it back

Explain to a friend how to solve x2 + 2x = 35 by factoring, why dividing both sides by x would be a mistake in an equation like x2 = 9x, and what you do with a solution that cannot fit a word problem.

Explaining it out loud shows what you really know. Did you…

  • I moved the 35 to the left to get x2 + 2x − 35 = 0, and said the equation must equal 0 before using the zero-product property.
  • I factored with the pair 7 and −5: (x + 7)(x − 5) = 0. Then I set each factor to 0 and got x = −7 or x = 5.
  • I checked both in the original equation: 49 − 14 = 35 and 25 + 10 = 35.
  • I said that dividing both sides by x assumes x ≠ 0 and loses the solution x = 0, so I should factor out x instead.
  • I said that a solution which does not fit a story, such as a negative length, is rejected, and I gave the reason.

Your 3-week memory plan

  1. TodayRead the lesson, then try the flashcards.
  2. TomorrowRecall it from memory first, then check.
  3. In 3 daysTake the true/false and practice questions.
  4. In 1 weekMix it with older topics and teach it back.
  5. In 3 weeksA last quick review before it fades.

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