2.1.20 · D5Algebra — Introduction & Intermediate
Question bank — Formation of quadratic with given roots
This is a rapid-fire trap bank for Formation of Quadratic with Given Roots. Every item below targets a thinking mistake, not arithmetic. Cover the answer, commit out loud, then reveal. If you can justify your answer (not just say "true"), you own the concept.
Two visual facts drive almost every trap below. First, the constant only stretches or flips the curve vertically — it never moves the roots (figure s02). Second, the roots are the anchor points on the x-axis, and their sum/product tell you where those anchors sit (figure s03).
True or false — justify
True or false: There is exactly one quadratic equation with roots and .
False — there are infinitely many: any with has those same roots, since multiplying an equation by a non-zero constant never changes its solutions (figure s02 shows several curves pinned to the same two crossings).
True or false: If is a root, then is a factor.
False — the factor is ; it must vanish at the root, and while .
True or false: Two different quadratics can have identical roots but different graphs.
True — they share x-intercepts but the constant stretches, compresses, or flips them vertically, so their vertices and steepness differ (this is exactly a vertical scaling from Graph transformations; see figure s02).
True or false: A quadratic with real coefficients can have roots and .
False — non-real roots of a real-coefficient quadratic must come in conjugate pairs, so if is a root then must be the other, leaving no room for (this pairing is why the parabola never crosses the axis yet still has real coefficients — see Complex roots).
True or false: If sum of roots is and product is , the equation is always .
Only when the leading coefficient is ; the full family is , but is the standard monic representative.
True or false: The roots of are , and so are the roots of .
True — the second equation is exactly times the first, so they have the same solution set; scaling by a non-zero constant preserves roots.
True or false: If a quadratic has a repeated root , it can be written .
True — a repeated root means the single factor appears twice, and .
True or false: Knowing only the sum of the roots is enough to form the quadratic.
False — you also need the product (or a second condition); infinitely many pairs share a given sum, e.g. sum fits both and .
Spot the error
"Roots are and , so the equation is ." Where's the slip?
The signs are flipped; root gives factor and root gives , so it should be .
"Sum of roots , product , therefore ." What's wrong?
The sum is subtracted: , giving ; the wrong version has roots , not .
", so for the product is ." Fix it.
The product is , not ; forgetting to divide by the leading coefficient is the classic Vieta's formulas slip.
"Roots are and , so product ." Where's the error?
It's a difference of squares: , not ; the middle terms cancel and the last term is subtracted.
"Given roots and the equation must pass through , the answer is ; I don't need to check the point." Is that safe reasoning?
It happens to work here because already gives , but in general the point fixes and you must verify it rather than assume .
" has roots summing to because sum and ." Spot the error.
; the person forgot the extra minus that turns into .
"To build a quadratic with root , I only need one factor ." What's missing?
A quadratic needs two roots (counting multiplicity); a single factor is linear, so you need a second factor like to reach degree two.
Why questions
Why does the factor , and not , capture a root at ?
Because a root is a value where the expression is exactly zero — the exact spot the curve touches the axis — and only vanishes precisely at ; adding a constant lifts the curve so its crossing slides elsewhere.
Why must we allow the constant at all if roots already determine "the" equation?
Roots pin down where the parabola meets the x-axis but not how tall or which way up it is, so carries the missing vertical scale (and, when , the flip) that distinguishes graphs sharing the same intercepts — picture figure s02's fan of curves.
Why does the sum of roots come with a minus sign in ?
Visually, the roots sit at and on the axis, and the vertex lands halfway between them at ; algebraically the cross terms in give , so shifting the roots rightward (larger sum) pushes the whole term negative — the minus records that the coefficient tracks the roots' position, not their raw value.
Why can we form a real-coefficient quadratic from a complex root without ever writing in the answer?
Multiplying a conjugate pair produces , where the imaginary parts cancel; graphically this is a parabola that floats entirely above (or below) the axis, never crossing, yet has ordinary real coefficients (see Complex roots).
Why is forming a quadratic from roots the "inverse" of the Quadratic formula?
The quadratic formula takes coefficients and returns roots; formation takes roots and returns coefficients, so each undoes the other's direction of travel.
Why does dividing by leave no remainder when is a root?
By Factor theorem and Polynomial long division, the remainder of dividing by equals , which is precisely because is a root.
Edge cases
If both roots are equal, say , what does the graph do at ?
The parabola touches the x-axis instead of crossing it, because the repeated factor never changes sign near .
What quadratic has roots and , and what's unusual about its coefficients?
It is ; both the linear and constant terms vanish since sum and product , giving the barest possible quadratic.
Can a genuine quadratic have exactly one root and no second root at all?
No — over the complex numbers every quadratic has two roots counting multiplicity; a "single visible root" is really a repeated root (multiplicity two).
If a quadratic's product of roots is , what does that force about at least one root?
At least one root must be , because a product equals zero only when a factor is zero, so the constant term is also .
What does a negative scaling constant do to the graph, and does it change the roots?
It flips the parabola upside-down (opens downward instead of upward) while keeping the exact same x-crossings, since multiplying by a negative number leaves the zeros untouched — figure s02 includes one such flipped curve through the same two roots.
What happens to the "form from sum and product" method if the product is negative?
Nothing breaks — a negative product just means the roots have opposite signs, e.g. sum , product gives with roots and .
Suppose someone gives you three points a parabola must pass through instead of roots — is the "" method still enough?
Not directly, because that form assumes you already know the roots; with three arbitrary points you'd solve for instead, since the intercepts aren't handed to you.
If you scale an equation by , do you still have a quadratic with those roots?
No — collapses everything to , which every satisfies, so it has no specific roots and isn't a quadratic at all; that's why we insist .
Recall One-line self-test
Root factor coefficients: state each arrow. ::: Root gives factor ; multiplying factors gives ; matching gives , .