Section 1
Solving Quadratic Equations by the Square Root Property
An equation in the form $a x^{2}+b x+c=0,$ where $a, b,$ and $c$ are real numbers and $a \neq 0,$ is $\mathrm{a}(\mathrm{n})$ _________ equation, also called a(n) ________ degree equation. The greatest degree of the variable is _____.
Which of the following are quadratic equations?A. $x+2 y=0$B. $x^{2}-8 x+16=0$C. $2 x^{2}-5 x=3$D. $x^{3}+x^{2}+4=0$
Which quadratic equation is in standard form?A. $x^{2}=25$B. $3 x^{2}-x=4$C. $(x-5)^{2}=16$D. $x^{2}-x-2=0$
The equation $x^{2}=-9$ has solutions that (are/are not) real numbers. These solutions involve the imaginary unit $i,$ which is defined as $i=$ ________ Thus, $i^{2}=$ ____________ For any positive real number $a$, we have $\sqrt{-a}=$ __________.
Match each equation in Column I with the correct description of its solution in Column II.I(a) $x^{2}=12$(b) $x^{2}=-16$(c) $x^{2}=\frac{25}{36}$(d) $x^{2}=100$IIA. Two nonreal complexB. Two integer solutions solutionsC. Two irrational solutionsD. Two rational solutions that are not integers
A student incorrectly solved the following equation as shown.$$\begin{array}{r}x^{2}-x-2=4 \\(x-2)(x+1)=4 \\x-2=4 \text { or } x+1=4 \\x=6 \quad \text { or } \quad x=3\end{array}$$Solve correctly and give the solution set.
A student solving $x^{2}=81$ wrote the solution set incorrectly as $\{9\} .$ Her teacher did not give her full credit. The student argued that because $9^{2}=81,$ her answer had to be correct. Give the correct solution set.
When solving a quadratic equation, a student obtained the solutions $x=\frac{3+2 \sqrt{5}}{2}$ or $x=\frac{3-2 \sqrt{5}}{2}$ and he wrote the solution set incorrectly as $\{3 \pm \sqrt{5}\}$. Give the correct solution set.
Solve using the zero-factor property.$$x^{2}-x-56=0$$
Solve using the zero-factor property.$$x^{2}-2 x-99=0$$
Solve using the zero-factor property.$$x^{2}-8 x+15=0$$
Solve using the zero-factor property.$$x^{2}-6 x+5=0$$
Solve using the zero-factor property.$$x^{2}=121$$
Solve using the zero-factor property.$$x^{2}=144$$
Solve using the zero-factor property.$$x^{2}-169=0$$
Solve using the zero-factor property.$$x^{2}-400=0$$
Solve using the zero-factor property.$$3 x^{2}-13 x=30$$
Solve using the zero-factor property.$$5 x^{2}-14 x=3$$
Solve using the zero-factor property.$$6 x^{2}+19 x+10=0$$
Solve using the zero-factor property.$$8 x^{2}+18 x+9=0$$
Solve using the square root property. Simplify all radicals.$$x^{2}=81$$
Solve using the square root property. Simplify all radicals.$$z^{2}=169$$
Solve using the square root property. Simplify all radicals.$$x^{2}=144$$
Solve using the square root property. Simplify all radicals.$$m^{2}=36$$
Solve using the square root property. Simplify all radicals.$$k^{2}=14$$
Solve using the square root property. Simplify all radicals.$$m^{2}=22$$
Solve using the square root property. Simplify all radicals.$$t^{2}=48$$
Solve using the square root property. Simplify all radicals.$$x^{2}=54$$
Solve using the square root property. Simplify all radicals.$$x^{2}=\frac{25}{4}$$
Solve using the square root property. Simplify all radicals.$$m^{2}=\frac{36}{121}$$
Solve using the square root property. Simplify all radicals.$$x^{2}=0.25$$
Solve using the square root property. Simplify all radicals.$$w^{2}=0.49$$
Solve using the square root property. Simplify all radicals.$$x^{2}-64=0$$
Solve using the square root property. Simplify all radicals.$$x^{2}-100=0$$
Solve using the square root property. Simplify all radicals.$$r^{2}-3=0$$
Solve using the square root property. Simplify all radicals.$$x^{2}-13=0$$
Solve using the square root property. Simplify all radicals.$$x^{2}-12=0$$
Solve using the square root property. Simplify all radicals.$$x^{2}-8=0$$
Solve using the square root property. Simplify all radicals.$$4 x^{2}-72=0$$
Solve using the square root property. Simplify all radicals.$$2 x^{2}-80=0$$
Solve using the square root property. Simplify all radicals.$$2 t^{2}+7=61$$
Solve using the square root property. Simplify all radicals.$$3 x^{2}+8=80$$
Solve using the square root property. Simplify all radicals.$$3 x^{2}-8=64$$
Solve using the square root property. Simplify all radicals.$$2 x^{2}-5=35$$
Solve using the square root property. Simplify all radicals.$$7 x^{2}=4$$
Solve using the square root property. Simplify all radicals.$$2 x^{2}=9$$
Solve using the square root property. Simplify all radicals.$$5 x^{2}+4=8$$
Solve using the square root property. Simplify all radicals.$$7 p^{2}-5=11$$
Solve using the square root property. Simplify all radicals.$$(x-3)^{2}=25$$
Solve using the square root property. Simplify all radicals.$$(x-7)^{2}=16$$
Solve using the square root property. Simplify all radicals.$$(x-4)^{2}=3$$
Solve using the square root property. Simplify all radicals.$$(x+3)^{2}=11$$
Solve using the square root property. Simplify all radicals.$$(x-8)^{2}=27$$
Solve using the square root property. Simplify all radicals.$$(p-5)^{2}=40$$
Solve using the square root property. Simplify all radicals.$$(3 x+2)^{2}=49$$
Solve using the square root property. Simplify all radicals.$$(5 t+3)^{2}=36$$
Solve using the square root property. Simplify all radicals.$$(4 x-3)^{2}=9$$
Solve using the square root property. Simplify all radicals.$$(7 z-5)^{2}=25$$
Solve using the square root property. Simplify all radicals.$$(5-2 x)^{2}=30$$
Solve using the square root property. Simplify all radicals.$$(3-2 x)^{2}=70$$
Solve using the square root property. Simplify all radicals.$$(3 k+1)^{2}=18$$
Solve using the square root property. Simplify all radicals.$$(5 z+6)^{2}=75$$
Solve using the square root property. Simplify all radicals.$$\left(\frac{1}{2} x+5\right)^{2}=12$$
Solve using the square root property. Simplify all radicals.$$\left(\frac{1}{3} m+4\right)^{2}=27$$
Solve using the square root property. Simplify all radicals.$$\left(x-\frac{1}{8}\right)^{2}=\frac{1}{64}$$
Solve using the square root property. Simplify all radicals.$$\left(x-\frac{1}{9}\right)^{2}=\frac{1}{81}$$
Solve using the square root property. Simplify all radicals.$$\left(x-\frac{1}{3}\right)^{2}=\frac{4}{9}$$
Solve using the square root property. Simplify all radicals.$$\left(x-\frac{1}{5}\right)^{2}=\frac{16}{25}$$
Solve using the square root property. Simplify all radicals.$$\left(x+\frac{1}{4}\right)^{2}=\frac{3}{16}$$
Solve using the square root property. Simplify all radicals.$$\left(x+\frac{1}{7}\right)^{2}=\frac{11}{49}$$
Solve using the square root property. Simplify all radicals.$$(4 x-1)^{2}-48=0$$
Solve using the square root property. Simplify all radicals.$$(2 x-5)^{2}-180=0$$
Solve each equation. (All solutions are nonreal complex numbers.)$$x^{2}=-100$$
Solve each equation. (All solutions are nonreal complex numbers.)$$x^{2}=-64$$
Solve each equation. (All solutions are nonreal complex numbers.)$$x^{2}=-26$$
Solve each equation. (All solutions are nonreal complex numbers.)$$x^{2}=-21$$
Solve each equation. (All solutions are nonreal complex numbers.)$$x^{2}=-12$$
Solve each equation. (All solutions are nonreal complex numbers.)$$x^{2}=-18$$
Solve each equation. (All solutions are nonreal complex numbers.)$$(x+3)^{2}=-4$$
Solve each equation. (All solutions are nonreal complex numbers.)$$(x-5)^{2}=-36$$
Solve each equation. (All solutions are nonreal complex numbers.)$$(r-5)^{2}=-3$$
Solve each equation. (All solutions are nonreal complex numbers.)$$(t+6)^{2}=-5$$
Solve each equation. (All solutions are nonreal complex numbers.)$$(6 k-1)^{2}=-8$$
Solve each equation. (All solutions are nonreal complex numbers.)$$(4 m-7)^{2}=-27$$
Use Galileo's formula to solve each problem. Round answers to the nearest tenth.The sculpture of American presidents at Mount Rushmore National Memorial is $500 \mathrm{ft}$ above the valley floor. How long would it take a rock dropped from the top of the sculpture to fall to the ground? (Data from www.travelsd.com)
Use Galileo's formula to solve each problem. Round answers to the nearest tenth.The Gateway Arch in St. Louis, Missouri, is $630 \mathrm{ft}$ tall. How long would it take an object dropped from the top of the arch to fall to the ground? (Data from www.gatewayarch.com)