Parent guide · 6 min read
Quadratic equations for kids: why you set it equal to zero first
Quadratic equations are where many students first feel algebra stop being "do the same to both sides". Most of the pain comes from one habit: trying to solve before the equation is in the right shape. This guide explains the one idea behind factoring, the formula for when factoring will not work, and the checks that catch most errors.
Try this tonight
4 things you can do at home
Get zero on one side first
Rewrite the equation as something = 0 before doing anything else. Try x² + 5x = -6: add 6 to both sides to get x² + 5x + 6 = 0. The method only works from this shape.
Look for two numbers that multiply to the last term and add to the middle one
For x² + 5x + 6, which two numbers multiply to 6 and add to 5? 2 and 3. Write the factors as (x + 2)(x + 3). Listing the pairs that multiply to 6 (1 and 6, 2 and 3) makes this a short search.
Set each bracket to zero
If (x + 2)(x + 3) = 0, then x + 2 = 0 or x + 3 = 0. So x = -2 or x = -3. Two brackets give two answers, and it is worth writing both down.
Check both answers in the original
Put x = -2 into x² + 5x + 6: 4 - 10 + 6 = 0, correct. Put in x = -3: 9 - 15 + 6 = 0, correct. A two-line check is the quickest way to catch a sign error.
Why the "equals zero" step matters
The reason factoring works is that zero is special: if two numbers multiply to give 0, at least one of them is 0. That is not true of any other number. If (x - 1)(x - 2) = 6, you cannot say x - 1 = 6 or x - 2 = 6, because many pairs of numbers multiply to 6.
So the equation has to be rearranged first. Expanding gives x² - 3x + 2 = 6, subtracting 6 gives x² - 3x - 4 = 0, and this factors as (x - 4)(x + 1) = 0. The answers are x = 4 and x = -1. Checking x = 4: (3)(2) = 6, correct. Checking x = -1: (-2)(-3) = 6, correct.
A worked example: when the x² has a number in front
For 2x² + 3x - 2 = 0, the quick "two numbers" search needs one extra step. Multiply the first and last numbers: 2 × -2 = -4. Look for two numbers that multiply to -4 and add to 3: 4 and -1. Split the middle term: 2x² + 4x - x - 2 = 0. Group the pairs: 2x(x + 2) - 1(x + 2) = 0, which gives (2x - 1)(x + 2) = 0.
So 2x - 1 = 0 gives x = 1/2, and x + 2 = 0 gives x = -2. Check x = -2: 8 - 6 - 2 = 0, correct.
When factoring does not work: the quadratic formula
Not every quadratic factors neatly, and some have no whole-number factors at all. The quadratic formula always works. For ax² + bx + c = 0, the solutions are x = (-b ± √(b² - 4ac)) / 2a. For 2x² + 3x - 2 = 0, a = 2, b = 3 and c = -2, so b² - 4ac = 9 + 16 = 25, the square root is 5, and x = (-3 ± 5) / 4. That gives x = 1/2 or x = -2, the same answers as factoring.
The number under the square root, b² - 4ac, tells you in advance what to expect: positive means two answers, zero means one, and negative means no real answers. A negative result is not a mistake, it is the equation telling you its graph never crosses the x-axis.
The shortcut that goes wrong: dividing by x
For x² = 3x, many students divide both sides by x to get x = 3. This loses the other answer, x = 0. The safe route is to move everything across: x² - 3x = 0, factor out x to get x(x - 3) = 0, and read off x = 0 or x = 3.
The same applies to square roots. x² = 9 has two answers, x = 3 and x = -3, because both 3 × 3 and (-3) × (-3) equal 9. Writing x = ±3 keeps both.
Signs it’s time for outside help
When home help has done what it can
- Tries to factor an equation that is not set equal to zero.
- Writes only one answer when there should be two.
- Divides both sides by x and loses a solution.
- Guesses factors by trial and error without checking that they add to the middle number.
- Makes sign errors when expanding brackets, such as (x - 4)(x + 1).
- Was comfortable with linear equations but freezes at the first x².
Questions parents ask
FAQ
What age do students learn quadratic equations?
In the US Common Core, solving quadratic equations is part of high school algebra, typically in Algebra 1 or Algebra 2 (roughly ages 13–16). In the Australian Curriculum, factorising and solving quadratics is covered in the upper secondary years. The order varies by school.
Which is better, factoring or the quadratic formula?
Factoring is quicker when the numbers are small and the equation factors neatly. The formula always works, so it is the fallback. Students benefit from knowing both and from recognising which suits the equation in front of them.
Why are there two answers?
The graph of a quadratic is a U-shaped curve called a parabola, and it can cross the x-axis in two places. Each crossing is one solution. Some touch it once, and some never cross it at all.
Does my child need to learn completing the square?
Many courses teach it, because it explains where the formula comes from and is used when graphing parabolas. For solving equations, factoring and the formula cover most of what is asked, so check what your child’s own course or school requires.
Further reading
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