# 2.EE: Exercises for Quadratic Equations

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## Solving Quadratic Equations Using the Square Root Property

###### Exercise A

In the following exercises, solve using the Square Root Property.

1. $$y^{2}=144$$
2. $$n^{2}-80=0$$
3. $$4 a^{2}=100$$
4. $$2 b^{2}=72$$
5. $$r^{2}+32=0$$
6. $$t^{2}+18=0$$
7. $$\frac{2}{3} w^{2}-20=30$$
8. $$5 c^{2}+3=19$$

1. $$y=\pm 12$$

3. $$a=\pm 5$$

5. $$r=\pm 4 \sqrt{2} i$$

7. $$w=\pm 5 \sqrt{3}$$

###### Exercise B

In the following exercises, solve using the Square Root Property.

1. $$(p-5)^{2}+3=19$$
2. $$(u+1)^{2}=45$$
3. $$\left(x-\frac{1}{4}\right)^{2}=\frac{3}{16}$$
4. $$\left(y-\frac{2}{3}\right)^{2}=\frac{2}{9}$$
5. $$(n-4)^{2}-50=150$$
6. $$(4 c-1)^{2}=-18$$
7. $$n^{2}+10 n+25=12$$
8. $$64 a^{2}+48 a+9=81$$

1. $$p=-1,9$$

3. $$x=\frac{1}{4} \pm \frac{\sqrt{3}}{4}$$

5. $$n=4 \pm 10 \sqrt{2}$$

7. $$n=-5 \pm 2 \sqrt{3}$$

### Solving Quadratic Equations by Completing the Square

###### Exercise A

In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

1. $$x^{2}+22 x$$
2. $$m^{2}-8 m$$
3. $$a^{2}-3 a$$
4. $$b^{2}+13 b$$

1. $$(x+11)^{2}$$

3. $$\left(a-\frac{3}{2}\right)^{2}$$

###### Exercise B

In the following exercises, solve by completing the square.

1. $$d^{2}+14 d=-13$$
2. $$y^{2}-6 y=36$$
3. $$m^{2}+6 m=-109$$
4. $$t^{2}-12 t=-40$$
5. $$v^{2}-14 v=-31$$
6. $$w^{2}-20 w=100$$
7. $$m^{2}+10 m-4=-13$$
8. $$n^{2}-6 n+11=34$$
9. $$a^{2}=3 a+8$$
10. $$b^{2}=11 b-5$$
11. $$(u+8)(u+4)=14$$
12. $$(z-10)(z+2)=28$$

1. $$d=-13,-1$$

3. $$m=-3 \pm 10 i$$

5. $$v=7 \pm 3 \sqrt{2}$$

7. $$m=-9,-1$$

9. $$a=\frac{3}{2} \pm \frac{\sqrt{41}}{2}$$

11. $$u=-6 \pm 2 \sqrt{2}$$

### Solving Quadratic Equations of the Form $$ax^{2}+bx+c=0$$ by Completing the Square

###### Exercise A

In the following exercises, solve by completing the square.

1. $$3 p^{2}-18 p+15=15$$
2. $$5 q^{2}+70 q+20=0$$
3. $$4 y^{2}-6 y=4$$
4. $$2 x^{2}+2 x=4$$
5. $$3 c^{2}+2 c=9$$
6. $$4 d^{2}-2 d=8$$
7. $$2 x^{2}+6 x=-5$$
8. $$2 x^{2}+4 x=-5$$

1. $$p=0,6$$

3. $$y=-\frac{1}{2}, 2$$

5. $$c=-\frac{1}{3} \pm \frac{2 \sqrt{7}}{3}$$

7. $$x=\frac{3}{2} \pm \frac{1}{2} i$$

###### Exercise B

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

1. $$4 x^{2}-5 x+1=0$$
2. $$7 y^{2}+4 y-3=0$$
3. $$r^{2}-r-42=0$$
4. $$t^{2}+13 t+22=0$$
5. $$4 v^{2}+v-5=0$$
6. $$2 w^{2}+9 w+2=0$$
7. $$3 m^{2}+8 m+2=0$$
8. $$5 n^{2}+2 n-1=0$$
9. $$6 a^{2}-5 a+2=0$$
10. $$4 b^{2}-b+8=0$$
11. $$u(u-10)+3=0$$
12. $$5 z(z-2)=3$$
13. $$\frac{1}{8} p^{2}-\frac{1}{5} p=-\frac{1}{20}$$
14. $$\frac{2}{5} q^{2}+\frac{3}{10} q=\frac{1}{10}$$
15. $$4 c^{2}+4 c+1=0$$
16. $$9 d^{2}-12 d=-4$$

1. $$x=\frac{1}{4}, 1$$

3. $$r=-6,7$$

5. $$v=\frac{-1 \pm \sqrt{21}}{8}$$

7. $$m=\frac{-4 \pm \sqrt{10}}{3}$$

9. $$a=\frac{5}{12} \pm \frac{\sqrt{23}}{12} i$$

11. $$u=5 \pm \sqrt{21}$$

13. $$p=\frac{4 \pm \sqrt{5}}{5}$$

15. $$c=-\frac{1}{2}$$

###### Exercise C

In the following exercises, determine the number of solutions for each quadratic equation.

1. $$9 x^{2}-6 x+1=0$$
2. $$3 y^{2}-8 y+1=0$$
3. $$7 m^{2}+12 m+4=0$$
4. $$5 n^{2}-n+1=0$$
1. $$5 x^{2}-7 x-8=0$$
2. $$7 x^{2}-10 x+5=0$$
3. $$25 x^{2}-90 x+81=0$$
4. $$15 x^{2}-8 x+4=0$$

1.

1. $$1$$
2. $$2$$
3. $$2$$
4. $$2$$
###### Exercise D

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.

1. $$16 r^{2}-8 r+1=0$$
2. $$5 t^{2}-8 t+3=9$$
3. $$3(c+2)^{2}=15$$
1. $$4 d^{2}+10 d-5=21$$
2. $$25 x^{2}-60 x+36=0$$
3. $$6(5 v-7)^{2}=150$$

1.

1. Factor
3. Square Root

### Solving Equations in Quadratic Form

###### Exercise A

In the following exercises, solve.

1. $$x^{4}-14 x^{2}+24=0$$
2. $$x^{4}+4 x^{2}-32=0$$
3. $$4 x^{4}-5 x^{2}+1=0$$
4. $$(2 y+3)^{2}+3(2 y+3)-28=0$$
5. $$x+3 \sqrt{x}-28=0$$
6. $$6 x+5 \sqrt{x}-6=0$$
7. $$x^{\frac{2}{3}}-10 x^{\frac{1}{3}}+24=0$$
8. $$x+7 x^{\frac{1}{2}}+6=0$$
9. $$8 x^{-2}-2 x^{-1}-3=0$$

1. $$x=\pm \sqrt{2}, x=\pm 2 \sqrt{3}$$

3. $$x=\pm 1, x=\pm \frac{1}{2}$$

5. $$x=16$$

7. $$x=64, x=216$$

9. $$x=-2, x=\frac{4}{3}$$

This page titled 2.EE: Exercises for Quadratic Equations is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Katherine Skelton.