➕ Algebra 1 · Quadratics
Solving Quadratics by Factoring: fun ways and memory help
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
The damage chain
In a made-up game, the damage of a move is the product of three multipliers: power × combo × shield-break. If a move does 0 damage, at least one of the three multipliers must have been 0. But if a move does 12 damage, you learn very little about any one multiplier. It could be 3 × 4 × 1, or 2 × 6 × 1, or 12 × 1 × 1.
How it connects: This is the zero-product property: if a product is 0, at least one factor is 0. So (x − 1)(x + 10) = 0 forces x − 1 = 0 or x + 10 = 0. But a product that equals 12 tells you nothing about what either factor equals on its own, which is why the equation must equal 0 before you split it into factors. Limit: the property is about products. For a sum, a total of 0 does not force any term to be 0, since 3 + (−3) = 0.
A ball tossed up from the ground
In a simplified model with no air resistance, a ball tossed straight up from ground level at 48 feet per second has height h = −16t2 + 48t feet after t seconds. It is on the ground at the moment of the toss and again when it comes back down. To find those times, set h = 0: −16t2 + 48t = 0, which factors as −16t(t − 3) = 0.
How it connects: The factors give t = 0, the instant of the toss, and t = 3, the landing 3 seconds later. The solution t = 0 is a real moment in the story, not “nothing”, and dividing both sides by t would lose it. Check the landing: −16(9) + 48(3) = −144 + 144 = 0. Limit: a real toss starts from a hand above the ground, and air changes the flight, so this is a simplified model.
Don’t put the cart before the horse
The proverb means that you should do things in the right order.
How it connects: In solving by factoring, the order matters. The equation must have 0 on one side before you factor and set each factor equal to 0. If you split a product that equals some other number, you get wrong answers. Limit: the proverb says only that order matters, not which order. For a quadratic equation the order is: zero on one side, factor completely, set each factor to 0, solve, then check.
Don’t throw the baby out with the bathwater
The idiom warns you not to throw away something valuable while you are getting rid of something you do not want.
How it connects: In x2 = 8x, dividing both sides by x gets rid of the unwanted x2, but it throws away a valid solution, x = 0, along with it. Factoring keeps both: x(x − 8) = 0 gives x = 0 or x = 8. Limit: the idiom is general advice. In algebra the exact rule is that you may divide by an expression only if it cannot be 0. Dividing both sides by a nonzero number, such as 4, is safe.
The zero partner
Why did the other factors dread being grouped with 0? No matter how big they were, the product always came out to nothing.
How it connects: One factor that is 0 makes the whole product 0, no matter how big the other factors are. The zero-product property runs this backward: if a product is 0, at least one factor must be 0. Limit: only 0 behaves this way. A product of 12 does not point to any one factor, so the equation must equal 0 before you use the property.
Memory help
Zebras Find Safe Shady Corners
In this lesson’s trick, the first letters give the five steps in order: Zero on one side, Factor completely, Set each factor to 0, Solve each one, Check in the original equation. Say it as “Zebras Find Safe Shady Corners.” It is for a quadratic equation that factors with integers.
Zero on one side
“Zero on one side, then factor all the way; set each factor to zero, solve and check, hooray!” Use it for quadratics that factor with integers. If no factor pair works, the rhyme stops and a later method (completing the square or the quadratic formula) takes over.
Only zero splits
If A × B = 0, then A = 0 or B = 0. If A × B = 12, nothing like that is true: 3 × 4 = 12 and 2 × 6 = 12, and neither factor has to be 12. Never set factors equal to a number other than 0. Move everything to one side first.
Never divide by x to “simplify”
From x2 = 5x, dividing both sides by x gives x = 5 and loses x = 0. Instead write x2 − 5x = 0 and factor: x(x − 5) = 0. Dividing by a number that is not zero, such as 5, is fine. Dividing by x is not, because x might be 0.
Flip the sign
The factor (x + 9) gives the solution x = −9, and the factor (x − 2) gives x = 2. Set the factor equal to 0, then solve. Do not just copy the sign from the factor.
Label the crossings
Sketch a parabola and write each solution under the point where it meets the x-axis. For (x − 1)(x − 3) = 0, mark 1 and 3 on the axis and write the factors (x − 1) and (x − 3) beside them. Each factor is the one that equals 0 at its crossing. With a double root, the parabola touches the axis at one point, its vertex, and turns around.