CTJan27 Online JMSS - Solving Radical Equations

1

Solve the basic radical equation for $x$: $$\sqrt{x - 3} = 4$$

$x = 7$ $x = 19$ $x = 13$ $x = 16$

2

Solve the equation involving a coefficient outside the radical: $$4\sqrt{2x} = 8$$

$x = 1$ $x = 4$ $x = 2$ $x = 8$

3

Solve the equation requiring rearrangement before squaring: $$5 + \sqrt{x + 1} = 10$$

$x = 4$ $x = 26$ $x = 24$ $x = 9$

4

Solve the radical equation involving a cube root: $$\sqrt[3]{x - 5} = -2$$

$x = 1$ $x = 13$ $x = -3$ $x = 3$

5

Find the solution to the equation: $$1 = 7 - 2\sqrt{x}$$

$x = 3$ $x = 1$ $x = 6$ $x = 9$

6

Solve the equation $\sqrt{x} = x - 6$. Be sure to check for extraneous solutions.

$x = 4$ only $x = 9$ only $x = 4$ and $x = 9$ No solution

7

The potential solutions found when solving $\sqrt{2x - 5} = x - 4$ are $x=7$ and $x=3$. Which of these is the extraneous solution?

$x = 7$ (valid) $x = 3$ (extraneous) Both solutions are valid There are no real solutions

8

Solve the equation where the variable is outside the radical: $$x = \sqrt{12 - x}$$

$x = 4$ $x = 3$ $x = -4$ $x = 6$

9

Find the unique solution to the equation: $$\sqrt{1 - x} = x + 5$$

$x = -8$ $x = -3$ $x = 1$ No solution exists

10

How many distinct valid solutions does the equation $\sqrt{3x - 2} = x - 4$ have?

Zero One Two Infinitely many

11

When solving $\sqrt{x+4} = x - 2$, the algebraic process yields $x=5$ and $x=0$. Why is $x=0$ an extraneous solution?

It makes the argument inside the radical negative. The value $x=0$ is not a solution because substituting it back results in $2 = -2$. The value $x=0$ is extraneous because $5$ is the only acceptable integer solution. It makes both sides of the original equation equal zero.

12

The speed $S$ (in mph) of a car can be estimated using $S = \sqrt{20L}$, where $L$ is the length of the skid marks in feet. If a car's speed was $60$ mph, what length of skid marks ($L$) would be expected?

$L = 3$ feet $L = 120$ feet $L = 180$ feet $L = 360$ feet

13

The time $t$ (in seconds) it takes an object to fall $h$ meters is given by $t = 0.45\sqrt{h}$. If an object falls for $2.7$ seconds, what height $h$ did it fall from?

$h = 6$ meters $h = 9$ meters $h = 36$ meters $h = 42$ meters

14

The period $T$ of a pendulum is $T = 2\pi\sqrt{\frac{L}{g}}$. If $g \approx 9.8$ m/s$^2$ and a pendulum has a period of $4$ seconds (use $\pi \approx 3.14$), approximately what is its length $L$ in meters?

$1.0$ m $4.0$ m $2.5$ m $9.8$ m

15

If the relationship between pressure $P$ and temperature $T$ is given by $T = 2\sqrt{P} + 5$, what is the pressure $P$ when the temperature $T$ is $15$?

$P = 10$ $P = 25$ $P = 5$ $P = 100$

16

Solve for $x$: $$6\sqrt{x + 2} = 18$$

$x = 5$ $x = 1$ $x = 7$ $x = 16$

17

Solve the radical equation: $$10 - 5\sqrt{x - 1} = 0$$

$x = 2$ $x = 5$ $x = 3$ $x = 11$

18

Solve for $x$: $$\sqrt{3x + 1} = \sqrt{x + 9}$$

$x = 8$ $x = 4$ $x = -2$ $x = 1$

19

When solving $\sqrt{2x + 1} = x - 7$, squaring both sides yields potential solutions $x=12$ and $x=4$. Which is the valid solution?

$x = 4$ only $x = 12$ only Both $x=4$ and $x=12$ The equation has no real solutions

20

The distance $d$ (in miles) to the horizon from a height $h$ (in feet) is $d = 1.22\sqrt{h}$. If a ship captain observes the horizon $6.1$ miles away, what is the height $h$ of the observation deck?

$h = 5$ feet $h = 10$ feet $h = 25$ feet $h = 36$ feet

21

Solve the radical equation $\sqrt{3x - 5} - 2 = x - 7$.

$x = 10$ $x = 3$ $x = 15$ $x = 5$

22

Solve the equation involving a cube root: $2\sqrt[3]{4x + 1} + 5 = 11$.

$x = 6$ $x = \frac{13}{2}$ $x = \frac{7}{4}$ $x = \frac{31}{8}$

23

Determine the valid solution to the equation $2\sqrt{x + 5} = x + 2$.

$x = 4$ $x = 1$ $x = -4$ $x = 7$

24

Solving the equation $\sqrt{2x + 15} = x$ results in two potential algebraic solutions. Which value is the correct solution to the original radical equation?

$x = 5$ $x = -3$ $x = 0$ $x = 1$

25

The equation $x - \sqrt{x + 11} = 1$ leads to the quadratic equation $x^2 - 3x - 10 = 0$. Identify the extraneous solution.

$x = 5$ $x = -2$ $x = 14$ $x = 0$

26

Find the solution set for the equation $\sqrt{5x + 1} = x - 1$.

$\{7, 0\}$ $\{7\}$ $\{0\}$ $\{3, 6\}$

27

Solve the two-radical equation: $\sqrt{x + 6} = 2 + \sqrt{x - 2}$.

$x = 3$ $x = 6$ $x = 10$ $x = 5$

28

Solve the equation $3\sqrt{2x + 1} = x + 5$.

$x = 4$ $x = 1$ $x = 8$ No solution

29

The time $t$ (in seconds) it takes for an object to fall from a height $h$ (in meters) is approximated by the formula $t = \frac{\sqrt{2h}}{4}$. If an object takes $3$ seconds to reach the ground, what was its initial height?

$72\text{ meters}$ $36\text{ meters}$ $18\text{ meters}$ $144\text{ meters}$

30

Solve the equation $\sqrt{x^2 + 5x + 1} - x = 2$.

$x = 3$ $x = -1$ $x = 4$ $x = 5$

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