Question

solve the following. be sure to check for extraneous solutions!

1. $sqrt{3x+10}=8$
2. $8sqrt{10x}-7=9$
3. $sqrt{8x+3}=3$
4. $sqrt3{16x}-7=1$
5. $sqrt3{4x+2}-6=-10$
6. $sqrt{4x+1}=sqrt{x+10}$
7. $x=sqrt{16x+225}$
8. $sqrt{44-2x}=x-10$

Explanation:

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Problem 1: $\sqrt{3x+10}=8$

Step1: Square both sides

$(\sqrt{3x+10})^2=8^2$
$3x+10=64$

Step2: Isolate $3x$

$3x=64-10$
$3x=54$

Step3: Solve for $x$

$x=\frac{54}{3}=18$

Step4: Check solution

$\sqrt{3(18)+10}=\sqrt{54+10}=\sqrt{64}=8$, valid.

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Problem 2: $8\sqrt{10x}-7=9$

Step1: Isolate the radical term

$8\sqrt{10x}=9+7$
$8\sqrt{10x}=16$

Step2: Simplify the radical term

$\sqrt{10x}=\frac{16}{8}=2$

Step3: Square both sides

$(\sqrt{10x})^2=2^2$
$10x=4$

Step4: Solve for $x$

$x=\frac{4}{10}=\frac{2}{5}$

Step5: Check solution

$8\sqrt{10(\frac{2}{5})}-7=8\sqrt{4}-7=8(2)-7=16-7=9$, valid.

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Problem 3: $\sqrt{8x+3}=3$

Step1: Square both sides

$(\sqrt{8x+3})^2=3^2$
$8x+3=9$

Step2: Isolate $8x$

$8x=9-3=6$

Step3: Solve for $x$

$x=\frac{6}{8}=\frac{3}{4}$

Step4: Check solution

$\sqrt{8(\frac{3}{4})+3}=\sqrt{6+3}=\sqrt{9}=3$, valid.

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Problem 4: $\sqrt[3]{16x}-7=1$

Step1: Isolate the radical term

$\sqrt[3]{16x}=1+7=8$

Step2: Cube both sides

$(\sqrt[3]{16x})^3=8^3$
$16x=512$

Step3: Solve for $x$

$x=\frac{512}{16}=32$

Step4: Check solution

$\sqrt[3]{16(32)}-7=\sqrt[3]{512}-7=8-7=1$, valid.

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Problem 5: $\sqrt[3]{4x+2}-6=-10$

Step1: Isolate the radical term

$\sqrt[3]{4x+2}=-10+6=-4$

Step2: Cube both sides

$(\sqrt[3]{4x+2})^3=(-4)^3$
$4x+2=-64$

Step3: Isolate $4x$

$4x=-64-2=-66$

Step4: Solve for $x$

$x=\frac{-66}{4}=-\frac{33}{2}$

Step5: Check solution

$\sqrt[3]{4(-\frac{33}{2})+2}-6=\sqrt[3]{-66+2}-6=\sqrt[3]{-64}-6=-4-6=-10$, valid.

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Problem 6: $\sqrt{4x+1}=\sqrt{x+10}$

Step1: Square both sides

$(\sqrt{4x+1})^2=(\sqrt{x+10})^2$
$4x+1=x+10$

Step2: Isolate $x$ terms

$4x-x=10-1$
$3x=9$

Step3: Solve for $x$

$x=\frac{9}{3}=3$

Step4: Check solution

Left: $\sqrt{4(3)+1}=\sqrt{13}$, Right: $\sqrt{3+10}=\sqrt{13}$, valid.

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Problem 7: $x=\sqrt{16x+225}$

Step1: Square both sides

$x^2=(\sqrt{16x+225})^2$
$x^2=16x+225$

Step2: Rearrange to quadratic form

$x^2-16x-225=0$

Step3: Solve quadratic equation

Use quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, where $a=1, b=-16, c=-225$:

$$\begin{align*} x&=\frac{16\pm\sqrt{(-16)^2-4(1)(-225)}}{2(1)}\\ &=\frac{16\pm\sqrt{256+900}}{2}\\ &=\frac{16\pm\sqrt{1156}}{2}\\ &=\frac{16\pm34}{2} \end{align*}$$

$x=\frac{16+34}{2}=25$ or $x=\frac{16-34}{2}=-9$

Step4: Check solutions

• For $x=25$: $25=\sqrt{16(25)+225}=\sqrt{400+225}=\sqrt{625}=25$, valid.
• For $x=-9$: $-9=\sqrt{16(-9)+225}=\sqrt{-144+225}=\sqrt{81}=9$, $-9

eq9$, extraneous.

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Problem 8: $\sqrt{44-2x}=x-10$

Step1: Square both sides

$(\sqrt{44-2x})^2=(x-10)^2$
$44-2x=x^2-20x+100$

Step2: Rearrange to quadratic form

$x^2-18x+56=0$

Step3: Factor the quadratic

$(x-14)(x-4)=0$

Step4: Solve for $x$

$x=14$ or $x=4$

Step5: Check solutions

• For $x=14$: $\sqrt{44-2(14)}=\sqrt{44-28}=\sqrt{16}=4$, $14-10=4$, valid.
• For $x=4$: $\sqrt{44-2(4)}=\sqrt{44-8}=\sqrt{36}=6$, $4-10=-6$, $6

eq-6$, extraneous.

Answer:

1. $x=18$
2. $x=\frac{2}{5}$
3. $x=\frac{3}{4}$
4. $x=32$
5. $x=-\frac{33}{2}$
6. $x=3$
7. $x=25$ (extraneous solution $x=-9$ rejected)
8. $x=14$ (extraneous solution $x=4$ rejected)