Completing the square rewrites a quadratic so the variable appears inside one squared binomial. The method is systematic and works even when integer factoring is inconvenient.
Square-root principle
The term that completes a square
Take half of the linear coefficient and square that result. Do not confuse \(\left(\frac b2\right)^2\) with \(\frac{b^2}{2}\).
- Move the constant
Separate the variable terms from the constant.
- Normalize
If the leading coefficient is not \(1\), divide every term by it.
- Half and square
Calculate \(\left(\frac b2\right)^2\) using the normalized linear coefficient.
- Balance
Add that same value to both sides.
- Factor
Rewrite the left side as a perfect-square binomial.
- Square root
Take both square roots and isolate the variable.
Complete the square without changing the equation
A seven-step equation flow that normalizes the leading coefficient, adds the same square to both sides, and retains both square-root branches.
- Original\[2x^2-12x+5=0\]
Write the equation in standard form.
- Move the constant\[2x^2-12x=-5\]
Subtract the constant from both sides.
- Normalize\[x^2-6x=-\frac{5}{2}\]
Divide every term by the leading coefficient.
- Add the square\[x^2-6x+9=-\frac{5}{2}+9\]
Half of negative six is negative three; its square is nine.
- Perfect-square form\[(x-3)^2=\frac{13}{2}\]
Factor the left and simplify the right.
- Take both square roots\[x-3=\pm\frac{\sqrt{26}}{2}\]
The plus-or-minus sign preserves both solutions.
- Solutions\[x=3\pm\frac{\sqrt{26}}{2}\]
Both values satisfy the original equation.
Complete the square with an odd linear coefficient
Solve \(x^2+5x-1=0\).
- Move
\(x^2+5x=1\).
- Complete
Add \(\left(\frac52\right)^2=\frac{25}{4}\) to both sides.
- Factor
\(\left(x+\frac52\right)^2=\frac{29}{4}\).
- Square root
\(x+\frac52=\pm\frac{\sqrt{29}}{2}\).
Definition of equal polynomials
Compare coefficients in an identity
If \(3x^2+(k-2)x+7=3x^2+5x+7\) for every \(x\), find \(k\).
- Match linear coefficients
\(k-2=5\).
- Solve
\(k=7\).
Common mistakes
- Add \(\left(\frac b2\right)^2\), not \(\frac{b^2}{2}\) or \(\frac b2\).
- Add the completion term to both sides to preserve equality.
- Divide every term by \(a\) before completing the square when \(a\ne1\).
- Keep the \(\pm\) branch after taking square roots.
- Compare coefficients only when two polynomials are equal for every input.
Find the completion term
What term completes \(x^2-14x\) to a perfect square?
- \(-7\)
- \(7\)
- \(49\)
- \(196\)
Show answer and explanation
Answer: \(49\)
Half of \(-14\) is \(-7\), and \((-7)^2=49\).
Key takeaways
What to remember
- Normalize the squared-term coefficient before completing the square.
- Add \(\left(\frac b2\right)^2\) to both sides.
- The perfect-square binomial uses the sign of half the linear coefficient.
- Taking a square root while solving requires both \(\pm\) branches.
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.