Practice Makes Perfect
Complete the Square of a Binomial Expression
In the following exercises, complete the square to make a perfect square trinomial. Then, write the result as a binomial squared.
Example \(\PageIndex{43}\)
\(a^2+10a\)
- Answer
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\((a+5)^2\)
Example \(\PageIndex{44}\)
\(b^2+12b\)
Example \(\PageIndex{45}\)
\(m^2+18m\)
- Answer
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\((m+9)^2\)
Example \(\PageIndex{46}\)
\(n^2+16n\)
Example \(\PageIndex{47}\)
\(m^2−24m\)
- Answer
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\((m−12)^2\)
Example \(\PageIndex{48}\)
\(n^2−16n\)
Example \(\PageIndex{49}\)
\(p^2−22p\)
- Answer
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\((p−11)^2\)
Example \(\PageIndex{50}\)
\(q^2−6q\)
Example \(\PageIndex{51}\)
\(x^2−9x\)
- Answer
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\((x−\frac{9}{2})^2\)
Example \(\PageIndex{52}\)
\(y^2+11y\)
Example \(\PageIndex{53}\)
\(p^2−13p\)
- Answer
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\((p−16)^2\)
Example \(\PageIndex{54}\)
\(q^2+34q\)
Solve Quadratic Equations of the Form \(x^2+bx+c=0\) by Completing the Square
In the following exercises, solve by completing the square.
Example \(\PageIndex{55}\)
\(v^2+6v=40\)
- Answer
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\(v=−10\), \(v=4\)
Example \(\PageIndex{56}\)
\(w^2+8w=65\)
Example \(\PageIndex{57}\)
\(u^2+2u=3\)
- Answer
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\(u=−3\), \(u=1\)
Example \(\PageIndex{58}\)
\(z^2+12z=−11\)
Example \(\PageIndex{59}\)
\(c^2−12c=13\)
- Answer
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\(c=−1\), \(c=13\)
Example \(\PageIndex{60}\)
\(d^2−8d=9\)
Example \(\PageIndex{61}\)
\(x^2−20x=21\)
- Answer
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\(x=−1\), \(x=21\)
Example \(\PageIndex{62}\)
\(y^2−2y=8\)
Example \(\PageIndex{63}\)
\(m^2+4m=−44\)
- Answer
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no real solution
Example \(\PageIndex{64}\)
\(n^2−2n=−3\)
Example \(\PageIndex{65}\)
\(r^2+6r=−11\)
- Answer
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no real solution
Example \(\PageIndex{66}\)
\(t^2−14t=−50\)
Example \(\PageIndex{67}\)
\(a^2−10a=−5\)
- Answer
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\(a=5\pm2\sqrt{5}\)
Example \(\PageIndex{68}\)
\(b^2+6b=41\)
Example \(\PageIndex{69}\)
\(u^2−14u+12=−1\)
- Answer
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\(u=1\), \(u=13\)
Example \(\PageIndex{70}\)
\(z^2+2z−5=2\)
Example \(\PageIndex{71}\)
\(v^2=9v+2\)
- Answer
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\(v=\frac{9}{2}\pm\frac{\sqrt{89}}{2}\)
Example \(\PageIndex{72}\)
\(w^2=5w−1\)
Example \(\PageIndex{73}\)
\((x+6)(x−2)=9\)
- Answer
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\(x=−7\), \(x=3\)
Example \(\PageIndex{74}\)
\((y+9)(y+7)=79\)
Solve Quadratic Equations of the Form \(ax^2+bx+c=0\) by Completing the Square
In the following exercises, solve by completing the square.
Example \(\PageIndex{75}\)
\(3m^2+30m−27=6\)
- Answer
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\(m=−11\), \(m=1\)
Example \(\PageIndex{76}\)
\(2n^2+4n−26=0\)
Example \(\PageIndex{77}\)
\(2c^2+c=6\)
- Answer
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\(c=−2\), \(c=\frac{3}{2}\)
Example \(\PageIndex{78}\)
\(3d^2−4d=15\)
Example \(\PageIndex{79}\)
\(2p^2+7p=14\)
- Answer
-
\(p=−\frac{7}{4}\pm\frac{\sqrt{161}}{4}\)
Example \(\PageIndex{80}\)
\(3q^2−5q=9\)
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?