Card 1 of 20
EasyStudy card 1
Prompt
What term completes \(x^2+bx\) to a square?
Answer
\(\left(\frac b2\right)^2\).
MathChapter 11: Quadratic Functions
Flashcards
Solving Quadratic Equations by Completing the Square Flashcards
Twenty cards on the completion term, balance, normalization, roots, and identities.
Current card1 / 20
Session progress0 of 20 studied
Card 1 of 20. Answer hidden.
Card 2 of 20
EasyStudy card 2
Prompt
What perfect-square identity results?
Answer
\(x^2+bx+\left(\frac b2\right)^2=\left(x+\frac b2\right)^2\).
Card 3 of 20
EasyStudy card 3
Prompt
Why add the completion term to both sides?
Answer
To preserve equality.
Card 4 of 20
EasyStudy card 4
Prompt
What must happen first when \(a\ne1\)?
Answer
Divide every term by \(a\) so the squared-term coefficient becomes \(1\).
Card 5 of 20
EasyStudy card 5
Prompt
What does \(u^2=d\), with \(d>0\), imply?
Answer
\(u=\pm\sqrt d\).
Card 6 of 20
EasyStudy card 6
Prompt
What term completes \(x^2-10x\)?
Answer
\(25\).
Card 7 of 20
EasyStudy card 7
Prompt
What binomial square equals \(x^2-10x+25\)?
Answer
\((x-5)^2\).
Card 8 of 20
MediumStudy card 8
Prompt
What is the common \(\pm\) mistake?
Answer
Keeping only the positive square-root branch.
Card 9 of 20
MediumStudy card 9
Prompt
Why is \(b^2/2\) usually the wrong completion term?
Answer
The required term is \((b/2)^2=b^2/4\).
Card 10 of 20
MediumStudy card 10
Prompt
What is the balance trap?
Answer
Adding the completion term to only one side changes the equation.
Card 11 of 20
MediumStudy card 11
Prompt
When can polynomial coefficients be compared directly?
Answer
When the polynomials are equal for every input.
Card 12 of 20
MediumStudy card 12
Prompt
Does equality at one input prove equal polynomials?
Answer
No; coefficient matching requires an identity.
Card 13 of 20
MediumStudy card 13
Prompt
Complete \(x^2+6x\).
Answer
Add \(9\) to obtain \((x+3)^2\).
Card 14 of 20
MediumStudy card 14
Prompt
Complete \(x^2-14x\).
Answer
Add \(49\) to obtain \((x-7)^2\).
Card 15 of 20
MediumStudy card 15
Prompt
What should happen before applying a root restriction?
Answer
Find all algebraic roots, then filter by the stated condition.
Card 16 of 20
MediumStudy card 16
Prompt
How do you verify a completed-square transformation?
Answer
Expand the square and reverse the added value to recover the original equation.
Card 17 of 20
HardStudy card 17
Prompt
Solve \(x^2+4x=3\) by completing the square.
Answer
\(x=-2\pm\sqrt7\).
Card 18 of 20
HardStudy card 18
Prompt
Why normalize before halving the linear coefficient?
Answer
The completion identity assumes the squared-term coefficient is \(1\).
Card 19 of 20
HardStudy card 19
Prompt
If \(x^2+kx+16=(x+4)^2\), what is \(k\)?
Answer
\(k=8\).
Card 20 of 20
HardStudy card 20
Prompt
What four checks validate a completed-square solution?
Answer
Normalization, equal additions, correct square expansion, and substitution of both roots.
Deck complete
You rated every card. Revisit difficult cards or shuffle for another pass.
Keyboard: Space reveals, left/right arrows navigate, 1–4 rate, and S shuffles.