Question

question 6
solve the equation \\(\sqrt{8x + 16} + 1 = x + 2\\)

answer
\\(\oplus\\) additional solution \\(\circleddash\\) no solution
\\(x = \square\\) submit answer

Explanation:

Step1: Isolate the square root

Subtract 1 from both sides: $\sqrt{8x + 16} = x + 2 - 1 = x + 1$

Step2: Square both sides

Square both sides to eliminate the square root: $(\sqrt{8x + 16})^2 = (x + 1)^2$
Which simplifies to: $8x + 16 = x^2 + 2x + 1$

Step3: Rearrange into quadratic equation

Bring all terms to one side: $x^2 + 2x + 1 - 8x - 16 = 0$
Simplify: $x^2 - 6x - 15 = 0$ Wait, no, wait, let's recalculate: $x^2 + 2x + 1 -8x -16 = x^2 -6x -15$? Wait, no, 1 -16 is -15, 2x -8x is -6x. Wait, but maybe I made a mistake. Wait, original after squaring: 8x +16 = x² + 2x +1. Then subtract 8x and 16: 0 = x² + 2x +1 -8x -16 → x² -6x -15 =0? Wait, but maybe I should check again. Wait, no, let's do it again.

Wait, original equation: $\sqrt{8x +16} +1 = x +2$

Subtract 1: $\sqrt{8x +16} = x +1$

Square both sides: $8x +16 = (x +1)^2 = x² + 2x +1$

Then, $x² + 2x +1 -8x -16 = 0$ → $x² -6x -15 = 0$? Wait, but maybe I made a mistake in the sign. Wait, 8x +16 -x² -2x -1 =0 → -x² +6x +15=0 → x² -6x -15=0. Then use quadratic formula: x = [6 ± √(36 +60)] /2 = [6 ± √96]/2 = [6 ± 4√6]/2 = 3 ± 2√6. Now check for extraneous solutions.

First, check x = 3 + 2√6:

Left side: $\sqrt{8(3 + 2√6) +16} +1 = \sqrt{24 + 16√6 +16} +1 = \sqrt{40 + 16√6} +1$

Right side: (3 + 2√6) +2 = 5 + 2√6

Wait, maybe I made a mistake in squaring. Wait, let's try another approach. Wait, maybe I messed up the initial step. Wait, original equation: $\sqrt{8x +16} +1 = x +2$

Let's isolate the square root: $\sqrt{8x +16} = x +1$

Now, the expression under the square root must be non-negative: 8x +16 ≥0 → x ≥ -2

Also, the right side x +1 must be non-negative (since square root is non-negative) → x +1 ≥0 → x ≥ -1

So x must be ≥ -1.

Now, let's check x=3: Wait, maybe I made a mistake in the quadratic. Wait, let's plug x=3 into original equation:

Left: √(24 +16) +1 = √40 +1 ≈6.324 +1=7.324

Right: 3 +2=5. Not equal. Wait, x=5:

Left: √(40 +16)+1=√56 +1≈7.483 +1=8.483

Right:5 +2=7. Not equal. Wait, maybe I made a mistake in the quadratic. Wait, let's do the algebra again.

$\sqrt{8x +16} +1 = x +2$

Subtract 1: $\sqrt{8x +16} = x +1$

Square both sides: 8x +16 = x² + 2x +1

Then, x² +2x +1 -8x -16 = x² -6x -15 =0

Quadratic formula: x = [6 ± √(36 + 60)]/2 = [6 ± √96]/2 = [6 ± 4√6]/2 = 3 ± 2√6 ≈3 ±4.899

So x≈7.899 or x≈-1.899

Check x≈-1.899: it's less than -1, so violates x≥-1, so discard.

Check x≈7.899:

Left: √(8*7.899 +16) +1 = √(63.192 +16) +1 = √79.192 +1≈8.9 +1=9.9

Right:7.899 +2≈9.899≈10. Close, due to rounding. Wait, maybe exact value:

x=3 + 2√6

Left: √(8*(3 + 2√6) +16) +1 = √(24 + 16√6 +16) +1 = √(40 + 16√6) +1

Right: (3 + 2√6) +2 =5 + 2√6

Now, square the right side: (5 + 2√6)^2 =25 + 20√6 +24=49 +20√6

Square the left side (without the +1): (√(40 +16√6))^2=40 +16√6

Then, (left side -1)^2 = (√(40 +16√6))^2 =40 +16√6. Wait, no, left side is √(40 +16√6) +1, so square of left side is 40 +16√6 + 2√(40 +16√6) +1=41 +16√6 + 2√(40 +16√6)

Square of right side is (5 + 2√6)^2=49 +20√6

Set equal: 41 +16√6 + 2√(40 +16√6) =49 +20√6

Subtract 41 +16√6: 2√(40 +16√6)=8 +4√6

Divide by 2: √(40 +16√6)=4 +2√6

Square both sides:40 +16√6=16 +16√6 +24=40 +16√6. So it works. So x=3 + 2√6 is a solution. Wait, but maybe I made a mistake earlier. Wait, let's check x=3:

Wait, no, x=3 + 2√6≈3 +4.899≈7.899. Let's plug into original equation:

Left: √(8*7.899 +16) +1=√(63.192 +16)+1=√79.192 +1≈8.9 +1=9.9

Right:7.899 +2≈9.899≈10. Close, due to rounding. So it's correct.

Wait, but maybe the quadratic was wrong. Wait, let's try x=3:

Wait, no, x=3: left side √(24 +16)+1=√40 +1≈7.32, right side 5. Not equal. x=5: left √(40 +16)+1=√56 +1≈8.48, right 7. Not equal. x=7: left √(56 +16)+1=√72 +1≈8.485 +1=9.485, right 9. Close. x=7: left √(56 +16)=√72≈8.485 +1=9.485, right 7 +2=9. Not equal. Wait, x=7.899: right side is 7.899 +2=9.899, left side is √(8*7.899 +16)+1=√(63.192 +16)+1=√79.192 +1≈8.9 +1=9.9. So it's correct.

Wait, but maybe the initial quadratic was wrong. Wait, let's do the algebra again:

Original equation: √(8x +16) +1 =x +2

Subtract 1: √(8x +16)=x +1

Square both sides: 8x +16 =x² +2x +1

Bring all terms to left: -x² +6x +15=0 → x² -6x -15=0. Yes, that's correct. Then quadratic formula: x=(6±√(36 +60))/2=(6±√96)/2=(6±4√6)/2=3±2√6. So x=3+2√6≈7.899 and x=3-2√6≈-1.899. Now, x=3-2√6≈-1.899 is less than -1 (since 2√6≈4.899, 3-4.899≈-1.899), so it's less than -1, which violates the condition that x +1 ≥0 (since square root is non-negative, so right side x +1 must be non-negative). So x=3-2√6 is extraneous. So the only solution is x=3 + 2√6. Wait, but maybe I made a mistake in the sign when moving terms. Wait, let's check x=3 + 2√6:

Left side: √(8*(3 + 2√6) +16) +1 = √(24 + 16√6 +16) +1 = √(40 + 16√6) +1

Right side: (3 + 2√6) +2 =5 + 2√6

Now, let's square the left side: (√(40 + 16√6) +1)^2 =40 + 16√6 + 2√(40 + 16√6) +1=41 + 16√6 + 2√(40 + 16√6)

Square the right side: (5 + 2√6)^2=25 + 20√6 +24=49 + 20√6

Set equal: 41 + 16√6 + 2√(40 + 16√6) =49 + 20√6

Subtract 41 + 16√6: 2√(40 + 16√6)=8 + 4√6

Divide by 2: √(40 + 16√6)=4 + 2√6

Square both sides:40 + 16√6=16 + 16√6 +24=40 + 16√6. So it's correct. So x=3 + 2√6 is a valid solution.

Wait, but maybe the problem expects an integer? Did I make a mistake in the initial equation? Let me check the original equation again: √(8x +16) +1 =x +2. Let's factor 8x +16=8(x +2). So √(8(x +2)) +1 =x +2. Let y =x +2. Then equation becomes √(8y) +1 =y. So √(8y)=y -1. Then 8y=(y -1)^2=y² -2y +1. So y² -10y +1=0. Then y=(10±√(100 -4))/2=(10±√96)/2=5±2√6. Then x = y -2=5±2√6 -2=3±2√6. So same result. So x=3 + 2√6 or x=3 -2√6. Then x=3 -2√6≈-1.899, which is less than -1 (since y =x +2=3 -2√6 +2=5 -2√6≈5 -4.899≈0.101, which is positive. Wait, y =x +2, so when x=3 -2√6, y=5 -2√6≈0.101, which is positive. So the right side y -1≈-0.899, which is negative, but square root of 8y is positive, so √(8y)=y -1 would imply positive=negative, which is impossible. So y -1 must be non-negative, so y ≥1. So y=5 +2√6≈9.899≥1, and y=5 -2√6≈0.101<1, so y=5 -2√6 is invalid. So x=3 +2√6 is the solution.

So the solution is x=3 + 2√6. But maybe the problem has a typo, or I made a mistake. Wait, let's check x=3:

Wait, no, x=3: left side √(24 +16)+1=√40 +1≈7.32, right side 5. Not equal. x=5: left √(40 +16)+1=√56 +1≈8.48, right 7. Not equal. x=7: left √(56 +16)+1=√72 +1≈9.48, right 9. Close. x=8: left √(64 +16)+1=√80 +1≈9.94, right 10. Close. x=3 + 2√6≈7.899, which is close to 8. So the solution is x=3 + 2√6.

Answer:

$x = 3 + 2\sqrt{6}$ (or approximately $x \approx 7.899$)