Solving Radical Equations and Inequalities - Math

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Question

Solve the equation for .

$$\sqrt{4x+9}$-8=5$

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Answer

$\small $\sqrt{4x+9}$-8=5$

Add to both sides.

$\small $\sqrt{4x+9}$=13$

Square both sides.

$\small $($\sqrt{4x+9}$)^2$$=13^2$$

$\small 4x+9=169$

Isolate .

$\small 4x=160$

$\small x=40$

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Question

Solve the following radical expression:

$$\sqrt{3x+1}$+2=7$

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Answer

Begin by subtracting from each side of the equation:

$$\sqrt{3x+1}$+2=7$

$$\sqrt{3x+1}$=5$

Now, square the equation:

Solve the linear equation:

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Question

Solve the following radical expression:

$\sqrt{2x+1}$-2=$\sqrt{2x-7}$

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Answer

Begin by squaring both sides of the equation:

$\sqrt{2x+1}$-2=$\sqrt{2x-7}$

$(2x+1)-4$\sqrt{2x+1}$+4=2x-7$

Combine like terms:

$-4$\sqrt{2x+1}$=-12$

$$\sqrt{2x+1}$=3$

Once again, square both sides of the equation:

Solve the linear equation:

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Question

Solve the following radical expression:

$$\sqrt{4x+29}$=(x+6)$

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Answer

Begin by squaring both sides of the equation:

$$\sqrt{4x+29}$=(x+6)$

Now, combine like terms:

Factor the equation:

However, when plugging in the values, does not work. Therefore, there is only one solution:

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Question

Solve for :

$\sqrt[3]{y+1}=3$

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Answer

Begin by cubing both sides:

$\sqrt[3]{y+1}=3$

Now we can easily solve:

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Question

Solve the following radical expression:

$\sqrt{y+12}$+1=$\sqrt{y+21}$

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Answer

Begin by squaring both sides of the equation:

$\sqrt{y+12}$+1=$\sqrt{y+21}$

$(y+12)+2$\sqrt{y+12}$+1=y+21$

Now, combine like terms and simplify:

$2$\sqrt{y+12}$=8$

$$\sqrt{y+12}$=4$

Once again, take the square of both sides of the equation:

Solve the linear equation:

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Question

Solve the following radical expression:

$$\sqrt{5+2x}$=x-5$

Tap to reveal answer

Answer

Begin by taking the square of both sides:

$$\sqrt{5+2x}$=x-5$

Combine like terms:

Factor the equation and solve:

However, when plugging in the values, does not work. Therefore, there is only one solution:

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Question

Solve the following radical expression:

$$\sqrt{x+11}$-x=-9$

Tap to reveal answer

Answer

To solve the radical expression, begin by subtracting from each side of the equation:

$$\sqrt{x+11}$-x=-9$

$$\sqrt{x+11}$=x-9$

Now, square both sides of the equation:

Combine like terms:

Factor the expression and solve:

However, when plugged into the original equation, does not work because the radical cannot be negative. Therefore, there is only one solution:

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Question

Solve for :

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Answer

To solve for in the equation

Square both sides of the equation

$\rightarrow $($\sqrt{x^2$$+12x+36}$)^2$ $=(4)^2$$

$\rightarrow $x^2$+12x+36 =16$

Set the equation equal to by subtracting the constant from both sides of the equation.

$\rightarrow $(x^2$+12x+36) - 16 =(16) -16$

$\rightarrow $x^2$+12x+20 =0$

Factor to find the zeros:

$\rightarrow (x+10)(x+2) = 0$

This gives the solutions

$x = \left \{ -10, -2\right \}$ .

Verify that these work in the original equation by substituting them in for . This is especially important to do in equations involving radicals to ensure no imaginary numbers (square roots of negative numbers) are created.

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Question

Solve the equation for .

$$\sqrt{4x+9}$-8=5$

Tap to reveal answer

Answer

$\small $\sqrt{4x+9}$-8=5$

Add to both sides.

$\small $\sqrt{4x+9}$=13$

Square both sides.

$\small $($\sqrt{4x+9}$)^2$$=13^2$$

$\small 4x+9=169$

Isolate .

$\small 4x=160$

$\small x=40$

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Question

Solve the following radical expression:

$$\sqrt{3x+1}$+2=7$

Tap to reveal answer

Answer

Begin by subtracting from each side of the equation:

$$\sqrt{3x+1}$+2=7$

$$\sqrt{3x+1}$=5$

Now, square the equation:

Solve the linear equation:

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Question

Solve the following radical expression:

$\sqrt{2x+1}$-2=$\sqrt{2x-7}$

Tap to reveal answer

Answer

Begin by squaring both sides of the equation:

$\sqrt{2x+1}$-2=$\sqrt{2x-7}$

$(2x+1)-4$\sqrt{2x+1}$+4=2x-7$

Combine like terms:

$-4$\sqrt{2x+1}$=-12$

$$\sqrt{2x+1}$=3$

Once again, square both sides of the equation:

Solve the linear equation:

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Question

Solve the following radical expression:

$$\sqrt{4x+29}$=(x+6)$

Tap to reveal answer

Answer

Begin by squaring both sides of the equation:

$$\sqrt{4x+29}$=(x+6)$

Now, combine like terms:

Factor the equation:

However, when plugging in the values, does not work. Therefore, there is only one solution:

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Question

Solve for :

$\sqrt[3]{y+1}=3$

Tap to reveal answer

Answer

Begin by cubing both sides:

$\sqrt[3]{y+1}=3$

Now we can easily solve:

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Question

Solve the following radical expression:

$\sqrt{y+12}$+1=$\sqrt{y+21}$

Tap to reveal answer

Answer

Begin by squaring both sides of the equation:

$\sqrt{y+12}$+1=$\sqrt{y+21}$

$(y+12)+2$\sqrt{y+12}$+1=y+21$

Now, combine like terms and simplify:

$2$\sqrt{y+12}$=8$

$$\sqrt{y+12}$=4$

Once again, take the square of both sides of the equation:

Solve the linear equation:

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Question

Solve the following radical expression:

$$\sqrt{5+2x}$=x-5$

Tap to reveal answer

Answer

Begin by taking the square of both sides:

$$\sqrt{5+2x}$=x-5$

Combine like terms:

Factor the equation and solve:

However, when plugging in the values, does not work. Therefore, there is only one solution:

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Question

Solve the following radical expression:

$$\sqrt{x+11}$-x=-9$

Tap to reveal answer

Answer

To solve the radical expression, begin by subtracting from each side of the equation:

$$\sqrt{x+11}$-x=-9$

$$\sqrt{x+11}$=x-9$

Now, square both sides of the equation:

Combine like terms:

Factor the expression and solve:

However, when plugged into the original equation, does not work because the radical cannot be negative. Therefore, there is only one solution:

← Didn't Know|Knew It →

Question

Solve for :

Tap to reveal answer

Answer

To solve for in the equation

Square both sides of the equation

$\rightarrow $($\sqrt{x^2$$+12x+36}$)^2$ $=(4)^2$$

$\rightarrow $x^2$+12x+36 =16$

Set the equation equal to by subtracting the constant from both sides of the equation.

$\rightarrow $(x^2$+12x+36) - 16 =(16) -16$

$\rightarrow $x^2$+12x+20 =0$

Factor to find the zeros:

$\rightarrow (x+10)(x+2) = 0$

This gives the solutions

$x = \left \{ -10, -2\right \}$ .

Verify that these work in the original equation by substituting them in for . This is especially important to do in equations involving radicals to ensure no imaginary numbers (square roots of negative numbers) are created.

← Didn't Know|Knew It →

Question

Solve the equation for .

$$\sqrt{4x+9}$-8=5$

Tap to reveal answer

Answer

$\small $\sqrt{4x+9}$-8=5$

Add to both sides.

$\small $\sqrt{4x+9}$=13$

Square both sides.

$\small $($\sqrt{4x+9}$)^2$$=13^2$$

$\small 4x+9=169$

Isolate .

$\small 4x=160$

$\small x=40$

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Question

Solve the following radical expression:

$$\sqrt{3x+1}$+2=7$

Tap to reveal answer

Answer

Begin by subtracting from each side of the equation:

$$\sqrt{3x+1}$+2=7$

$$\sqrt{3x+1}$=5$

Now, square the equation:

Solve the linear equation:

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