# 10.3E: Exercises

• • OpenStax
• OpenStax
$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

## Practice Makes Perfect

In the following exercises, solve by using the Quadratic Formula.

##### Example $$\PageIndex{31}$$

$$4m^2+m−3=0$$

$$m=−1$$, $$m=\frac{3}{4}$$

##### Example $$\PageIndex{32}$$

$$4n^2−9n+5=0$$

##### Example $$\PageIndex{33}$$

$$2p^2−7p+3=0$$

$$p=\frac{1}{2}$$, $$p=3$$

##### Example $$\PageIndex{34}$$

$$3q^2+8q−3=0$$

##### Example $$\PageIndex{35}$$

$$p^2+7p+12=0$$

$$p=−4$$, $$p=−3$$

##### Example $$\PageIndex{36}$$

$$q^2+3q−18=0$$

##### Example $$\PageIndex{37}$$

$$r^2−8r−33=0$$

$$r=−3$$, $$r=11$$

##### Example $$\PageIndex{38}$$

$$t^2+13t+40=0$$

##### Example $$\PageIndex{39}$$

$$3u^2+7u−2=0$$

$$u=\frac{−7\pm\sqrt{73}}{6}$$

##### Example $$\PageIndex{40}$$

$$6z^2−9z+1=0$$

##### Example $$\PageIndex{41}$$

$$2a^2−6a+3=0$$

$$a=\frac{3\pm\sqrt{3}}{2}$$

##### Example $$\PageIndex{42}$$

$$5b^2+2b−4=0$$

##### Example $$\PageIndex{43}$$

$$2x^2+3x+9=0$$

no real solution

##### Example $$\PageIndex{44}$$

$$6y^2−5y+2=0$$​​​​​​

##### Example $$\PageIndex{45}$$

$$v(v+5)−10=0$$

$$v=\frac{−5\pm\sqrt{65}}{2}$$

##### Example $$\PageIndex{46}$$

$$3w(w−2)−8=0$$​​​​​​

##### Example $$\PageIndex{47}$$

$$\frac{1}{3}m^2+\frac{1}{12}m=\frac{1}{4}$$

$$m=−1$$, $$m=\frac{3}{4}$$

##### Example $$\PageIndex{48}$$

$$\frac{1}{3}n^2+n=−\frac{1}{2}$$

##### Example $$\PageIndex{49}$$

$$16c^2+24c+9=0$$

$$c=−\frac{3}{4}$$

##### Example $$\PageIndex{50}$$

$$25d^2−60d+36=0$$

##### Example $$\PageIndex{51}$$

5m^2+2m−7=0

$$m=−\frac{7}{5}$$, $$m=1$$

##### Example $$\PageIndex{52}$$

$$8n^2−3n+3=0$$​​​​​​

##### Example $$\PageIndex{53}$$

$$p^2−6p−27=0$$

$$p=−3$$, $$p=9$$

##### Example $$\PageIndex{54}$$

$$25q^2+30q+9=0$$

##### Example $$\PageIndex{55}$$

$$4r^2+3r−5=0$$

$$r=\frac{−3\pm\sqrt{89}}{8}$$​​​​​

##### Example $$\PageIndex{56}$$

$$3t(t−2)=2$$​​​​​​​

##### Example $$\PageIndex{57}$$

$$2a^2+12a+5=0$$

$$a=\frac{−6\pm\sqrt{26}}{2}$$​​​​​​​

##### Example $$\PageIndex{58}$$

$$4d^2−7d+2=0$$​​​​​​​

##### Example $$\PageIndex{59}$$

$$\frac{3}{4}b^2+\frac{1}{2}b=\frac{3}{8}$$

$$b=\frac{−2\pm\sqrt{11}}{6}$$​​​​​​​

##### Example $$\PageIndex{60}$$

$$\frac{1}{9}c^2+\frac{2}{3}c=3$$​​​​​​​

##### Example $$\PageIndex{61}$$

$$2x^2+12x−3=0$$

$$x=\frac{−6\pm\sqrt{42}}{4}$$​​​​​​​

##### Example $$\PageIndex{62}$$

$$16y^2+8y+1=0$$

​​​​​​​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.

##### Example $$\PageIndex{63}$$
1. $$4x^2−5x+16=0$$
2. $$36y^2+36y+9=0$$
3. $$6m^2+3m−5=0$$
4. $$18n^2−7n+3=0$$
1. no real solutions
2. 1
3. 2
4. no real solutions
##### Example $$\PageIndex{64}$$
1. $$9v^2−15v+25=0$$
2. $$100w^2+60w+9=0$$
3. $$5c^2+7c−10=0$$
4. $$15d^2−4d+8=0$$
##### Example $$\PageIndex{65}$$
1. $$r^2+12r+36=0$$
2. $$8t^2−11t+5=0$$
3. $$4u^2−12u+9=0$$
4. $$3v^2−5v−1=0$$
1. 1
2. no real solutions
3. 1
4. 2
##### Example $$\PageIndex{66}$$
1. $$25p^2+10p+1=0$$
2. $$7q^2−3q−6=0$$
3. $$7y^2+2y+8=0$$
4. $$25z^2−60z+36=0$$

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.

##### Example $$\PageIndex{67}$$
1. $$x^2−5x−24=0$$
2. $$(y+5)^2=12$$
3. $$14m^2+3m=11$$
1. factor
2. square root
##### Example $$\PageIndex{68}$$
1. $$(8v+3)^2=81$$
2. $$w^2−9w−22=0$$
3. $$4n^2−10=6$$​​​​​​​
##### Example $$\PageIndex{69}$$
1. $$6a^2+14=20$$
2. $$(x−\frac{1}{4})^2=\frac{5}{16}$$
3. $$y^2−2y=8$$
1. factor
2. square root
3. factor​​​​​​​
##### Example $$\PageIndex{70}$$
1. $$8b^2+15b=4$$
2. $$\frac{5}{9}v^2−\frac{2}{3}v=1$$
3. $$(w+\frac{4}{3})^2=\frac{2}{9}$$

## Everyday Math

##### Example $$\PageIndex{71}$$

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.

5 seconds, 8 seconds

##### Example $$\PageIndex{72}$$

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

##### Example $$\PageIndex{73}$$

Solve the equation $$x^2+10x=200$$

1. by completing the square
3. Which method do you prefer? Why?
1. −20, 10
2. −20, 10
##### Example $$\PageIndex{74}$$

Solve the equation $$12y^2+23y=24$$

1. by completing the square
3. 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?

This page titled 10.3E: Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.