10.3E: Exercises
- Page ID
- 30285
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Solve Quadratic Equations Using the Quadratic Formula
In the following exercises, solve by using the Quadratic Formula.
\(4m^2+m−3=0\)
- Answer
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\(m=−1\), \(m=\frac{3}{4}\)
\(4n^2−9n+5=0\)
\(2p^2−7p+3=0\)
- Answer
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\(p=\frac{1}{2}\), \(p=3\)
\(3q^2+8q−3=0\)
\(p^2+7p+12=0\)
- Answer
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\(p=−4\), \(p=−3\)
\(q^2+3q−18=0\)
\(r^2−8r−33=0\)
- Answer
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\(r=−3\), \(r=11\)
\(t^2+13t+40=0\)
\(3u^2+7u−2=0\)
- Answer
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\(u=\frac{−7\pm\sqrt{73}}{6}\)
\(6z^2−9z+1=0\)
\(2a^2−6a+3=0\)
- Answer
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\(a=\frac{3\pm\sqrt{3}}{2}\)
\(5b^2+2b−4=0\)
\(2x^2+3x+9=0\)
- Answer
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no real solution
\(6y^2−5y+2=0\)
\(v(v+5)−10=0\)
- Answer
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\(v=\frac{−5\pm\sqrt{65}}{2}\)
\(3w(w−2)−8=0\)
\(\frac{1}{3}m^2+\frac{1}{12}m=\frac{1}{4}\)
- Answer
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\(m=−1\), \(m=\frac{3}{4}\)
\(\frac{1}{3}n^2+n=−\frac{1}{2}\)
\(16c^2+24c+9=0\)
- Answer
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\(c=−\frac{3}{4}\)
\(25d^2−60d+36=0\)
5m^2+2m−7=0
- Answer
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\(m=−\frac{7}{5}\), \(m=1\)
\(8n^2−3n+3=0\)
\(p^2−6p−27=0\)
- Answer
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\(p=−3\), \(p=9\)
\(25q^2+30q+9=0\)
\(4r^2+3r−5=0\)
- Answer
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\(r=\frac{−3\pm\sqrt{89}}{8}\)
\(3t(t−2)=2\)
\(2a^2+12a+5=0\)
- Answer
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\(a=\frac{−6\pm\sqrt{26}}{2}\)
\(4d^2−7d+2=0\)
\(\frac{3}{4}b^2+\frac{1}{2}b=\frac{3}{8}\)
- Answer
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\(b=\frac{−2\pm\sqrt{11}}{6}\)
\(\frac{1}{9}c^2+\frac{2}{3}c=3\)
\(2x^2+12x−3=0\)
- Answer
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\(x=\frac{−6\pm\sqrt{42}}{4}\)
\(16y^2+8y+1=0\)
Use the Discriminant to Predict the Number of Solutions of a Quadratic Equation
In the following exercises, determine the number of solutions to each quadratic equation.
- \(4x^2−5x+16=0\)
- \(36y^2+36y+9=0\)
- \(6m^2+3m−5=0\)
- \(18n^2−7n+3=0\)
- Answer
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- no real solutions
- 1
- 2
- no real solutions
- \(9v^2−15v+25=0\)
- \(100w^2+60w+9=0\)
- \(5c^2+7c−10=0\)
- \(15d^2−4d+8=0\)
- \(r^2+12r+36=0\)
- \(8t^2−11t+5=0\)
- \(4u^2−12u+9=0\)
- \(3v^2−5v−1=0\)
- Answer
-
- 1
- no real solutions
- 1
- 2
- \(25p^2+10p+1=0\)
- \(7q^2−3q−6=0\)
- \(7y^2+2y+8=0\)
- \(25z^2−60z+36=0\)
Identify the Most Appropriate Method to Use to Solve a Quadratic Equation
In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.
- \(x^2−5x−24=0\)
- \((y+5)^2=12\)
- \(14m^2+3m=11\)
- Answer
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- factor
- square root
- Quadratic Formula
- \((8v+3)^2=81\)
- \(w^2−9w−22=0\)
- \(4n^2−10=6\)
- \(6a^2+14=20\)
- \((x−\frac{1}{4})^2=\frac{5}{16}\)
- \(y^2−2y=8\)
- Answer
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- factor
- square root
- factor
- \(8b^2+15b=4\)
- \(\frac{5}{9}v^2−\frac{2}{3}v=1\)
- \((w+\frac{4}{3})^2=\frac{2}{9}\)
Everyday Math
A flare is fired straight up from a ship at sea. Solve the equation \(16(t^2−13t+40)=0\) for t, the number of seconds it will take for the flare to be at an altitude of 640 feet.
- Answer
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5 seconds, 8 seconds
An architect is designing a hotel lobby. She wants to have a triangular window looking out to an atrium, with the width of the window 6 feet more than the height. Due to energy restrictions, the area of the window must be 140 square feet. Solve the equation \(\frac{1}{2}h^2+3h=140\) for h, the height of the window.
Writing Exercises
Solve the equation \(x^2+10x=200\)
- by completing the square
- using the Quadratic Formula
- Which method do you prefer? Why?
- Answer
-
- −20, 10
- −20, 10
- answers will vary
Solve the equation \(12y^2+23y=24\)
- by completing the square
- using the Quadratic Formula
- Which method do you prefer? Why?
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?