Answer :
To solve the given equation \( 25x^2 + 64 = 289 \), follow these detailed steps:
1. Isolate the quadratic term:
[tex]\[ 25x^2 + 64 = 289 \][/tex]
Subtract 64 from both sides of the equation to isolate the \( 25x^2 \) term:
[tex]\[ 25x^2 = 289 - 64 \][/tex]
This simplifies to:
[tex]\[ 25x^2 = 225 \][/tex]
2. Solve for \( x^2 \):
Divide both sides of the equation by 25:
[tex]\[ x^2 = \frac{225}{25} \][/tex]
Simplify the right side:
[tex]\[ x^2 = 9 \][/tex]
3. Take the square root of both sides:
To solve for \( x \), take the square root of both sides of the equation:
[tex]\[ x = \pm \sqrt{9} \][/tex]
Since the square root of 9 is 3, we have:
[tex]\[ x = \pm 3 \][/tex]
Thus, the solutions to the equation are:
[tex]\[ x = -3 \quad \text{and} \quad x = 3 \][/tex]
Comparing these solutions with the given multiple-choice options, we find that:
A. \( x = -3, 3 \) is correct.
B. \( x = -3\sqrt{3}, 3\sqrt{3} \) is incorrect.
C. \( x = -9, 9 \) is incorrect.
D. \( x = -\sqrt{3}, \sqrt{3} \) is incorrect.
Therefore, the correct answer is:
[tex]\[ \boxed{A. \ x = -3, 3} \][/tex]
1. Isolate the quadratic term:
[tex]\[ 25x^2 + 64 = 289 \][/tex]
Subtract 64 from both sides of the equation to isolate the \( 25x^2 \) term:
[tex]\[ 25x^2 = 289 - 64 \][/tex]
This simplifies to:
[tex]\[ 25x^2 = 225 \][/tex]
2. Solve for \( x^2 \):
Divide both sides of the equation by 25:
[tex]\[ x^2 = \frac{225}{25} \][/tex]
Simplify the right side:
[tex]\[ x^2 = 9 \][/tex]
3. Take the square root of both sides:
To solve for \( x \), take the square root of both sides of the equation:
[tex]\[ x = \pm \sqrt{9} \][/tex]
Since the square root of 9 is 3, we have:
[tex]\[ x = \pm 3 \][/tex]
Thus, the solutions to the equation are:
[tex]\[ x = -3 \quad \text{and} \quad x = 3 \][/tex]
Comparing these solutions with the given multiple-choice options, we find that:
A. \( x = -3, 3 \) is correct.
B. \( x = -3\sqrt{3}, 3\sqrt{3} \) is incorrect.
C. \( x = -9, 9 \) is incorrect.
D. \( x = -\sqrt{3}, \sqrt{3} \) is incorrect.
Therefore, the correct answer is:
[tex]\[ \boxed{A. \ x = -3, 3} \][/tex]
