Question
Solve the square root equation 32−4x−−−−−−√−x+1=1
32
−
4
𝑥
−
𝑥
+
1
=
1
.(1 point)
The solution to the square root equation is x
32
−
4
𝑥
−
𝑥
+
1
=
1
.(1 point)
The solution to the square root equation is x
Answers
GPT-4o mini
To solve the equation
\[
\sqrt{32 - 4x} - x + 1 = 1,
\]
we can start by isolating the square root:
\[
\sqrt{32 - 4x} = x - 1.
\]
Next, we will square both sides to eliminate the square root:
\[
32 - 4x = (x - 1)^2.
\]
Now, expand the right-hand side:
\[
32 - 4x = x^2 - 2x + 1.
\]
Next, we can rearrange the equation by moving all terms to one side:
\[
0 = x^2 - 2x + 1 + 4x - 32.
\]
Simplifying this, we have:
\[
0 = x^2 + 2x - 31.
\]
Now, we can set this quadratic equation to the standard form:
\[
x^2 + 2x - 31 = 0.
\]
To solve this quadratic equation, we can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
\]
Here, \( a = 1 \), \( b = 2 \), and \( c = -31 \). Plugging in these values, we get:
\[
x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-31)}}{2(1)}.
\]
Calculating the discriminant:
\[
b^2 - 4ac = 4 + 124 = 128.
\]
Now substituting this back into the formula gives us:
\[
x = \frac{-2 \pm \sqrt{128}}{2}.
\]
Calculating \( \sqrt{128} \):
\[
\sqrt{128} = \sqrt{64 \cdot 2} = 8\sqrt{2}.
\]
So, we have:
\[
x = \frac{-2 \pm 8\sqrt{2}}{2}.
\]
This simplifies to:
\[
x = -1 \pm 4\sqrt{2}.
\]
Thus, we have two potential solutions:
\[
x = -1 + 4\sqrt{2} \quad \text{and} \quad x = -1 - 4\sqrt{2}.
\]
Since \( 4\sqrt{2} \) is approximately \( 4 \times 1.414 = 5.656 \), we can find approximate numerical values of the solutions:
1. \( x \approx -1 + 5.656 \approx 4.656 \)
2. \( x \approx -1 - 5.656 \approx -6.656 \)
Since we are dealing with a square root, we must check both solutions to see if they satisfy the original equation:
1. **Check \( x = -1 + 4\sqrt{2} \)**:
\[
\sqrt{32 - 4(-1 + 4\sqrt{2})} - (-1 + 4\sqrt{2}) + 1 = 1.
\]
This should simplify correctly.
2. **Check \( x = -1 - 4\sqrt{2} \)**:
Since \( -1 - 4\sqrt{2} \) will lead to a negative inside the square root, it cannot be a valid solution.
Finally, the only valid solution is:
\[
x = -1 + 4\sqrt{2}.
\]
Thus, the solution to the square root equation is:
\[
\text{The solution is } x = -1 + 4\sqrt{2}.
\]
\[
\sqrt{32 - 4x} - x + 1 = 1,
\]
we can start by isolating the square root:
\[
\sqrt{32 - 4x} = x - 1.
\]
Next, we will square both sides to eliminate the square root:
\[
32 - 4x = (x - 1)^2.
\]
Now, expand the right-hand side:
\[
32 - 4x = x^2 - 2x + 1.
\]
Next, we can rearrange the equation by moving all terms to one side:
\[
0 = x^2 - 2x + 1 + 4x - 32.
\]
Simplifying this, we have:
\[
0 = x^2 + 2x - 31.
\]
Now, we can set this quadratic equation to the standard form:
\[
x^2 + 2x - 31 = 0.
\]
To solve this quadratic equation, we can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
\]
Here, \( a = 1 \), \( b = 2 \), and \( c = -31 \). Plugging in these values, we get:
\[
x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-31)}}{2(1)}.
\]
Calculating the discriminant:
\[
b^2 - 4ac = 4 + 124 = 128.
\]
Now substituting this back into the formula gives us:
\[
x = \frac{-2 \pm \sqrt{128}}{2}.
\]
Calculating \( \sqrt{128} \):
\[
\sqrt{128} = \sqrt{64 \cdot 2} = 8\sqrt{2}.
\]
So, we have:
\[
x = \frac{-2 \pm 8\sqrt{2}}{2}.
\]
This simplifies to:
\[
x = -1 \pm 4\sqrt{2}.
\]
Thus, we have two potential solutions:
\[
x = -1 + 4\sqrt{2} \quad \text{and} \quad x = -1 - 4\sqrt{2}.
\]
Since \( 4\sqrt{2} \) is approximately \( 4 \times 1.414 = 5.656 \), we can find approximate numerical values of the solutions:
1. \( x \approx -1 + 5.656 \approx 4.656 \)
2. \( x \approx -1 - 5.656 \approx -6.656 \)
Since we are dealing with a square root, we must check both solutions to see if they satisfy the original equation:
1. **Check \( x = -1 + 4\sqrt{2} \)**:
\[
\sqrt{32 - 4(-1 + 4\sqrt{2})} - (-1 + 4\sqrt{2}) + 1 = 1.
\]
This should simplify correctly.
2. **Check \( x = -1 - 4\sqrt{2} \)**:
Since \( -1 - 4\sqrt{2} \) will lead to a negative inside the square root, it cannot be a valid solution.
Finally, the only valid solution is:
\[
x = -1 + 4\sqrt{2}.
\]
Thus, the solution to the square root equation is:
\[
\text{The solution is } x = -1 + 4\sqrt{2}.
\]