Asked by T-Swizzle
Solve the following quadratic equation using the perfect square trinomial pattern: x^2 - 16x + 64 = 0
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Answered by
GPT-4o mini
To solve the quadratic equation \( x^2 - 16x + 64 = 0 \) using the perfect square trinomial pattern, we first identify if the equation can be expressed as a perfect square.
The general form of a perfect square trinomial is:
\[
(a - b)^2 = a^2 - 2ab + b^2
\]
In our equation:
- The first term, \( x^2 \), corresponds to \( a^2 \) where \( a = x \).
- The constant term \( 64 \) corresponds to \( b^2 \), hence \( b = 8 \) since \( 8^2 = 64 \).
- The middle term \( -16x \) corresponds to \( -2ab \), so we can verify this:
\[
-2ab = -2(x)(8) = -16x
\]
This confirms the equation can be expressed as a perfect square. Therefore, we rewrite the equation:
\[
(x - 8)^2 = 0
\]
Next, we solve for \( x \) by taking the square root of both sides:
\[
x - 8 = 0
\]
Thus,
\[
x = 8
\]
Finally, we conclude that the solution to the quadratic equation \( x^2 - 16x + 64 = 0 \) is:
\[
\boxed{8}
\]
The general form of a perfect square trinomial is:
\[
(a - b)^2 = a^2 - 2ab + b^2
\]
In our equation:
- The first term, \( x^2 \), corresponds to \( a^2 \) where \( a = x \).
- The constant term \( 64 \) corresponds to \( b^2 \), hence \( b = 8 \) since \( 8^2 = 64 \).
- The middle term \( -16x \) corresponds to \( -2ab \), so we can verify this:
\[
-2ab = -2(x)(8) = -16x
\]
This confirms the equation can be expressed as a perfect square. Therefore, we rewrite the equation:
\[
(x - 8)^2 = 0
\]
Next, we solve for \( x \) by taking the square root of both sides:
\[
x - 8 = 0
\]
Thus,
\[
x = 8
\]
Finally, we conclude that the solution to the quadratic equation \( x^2 - 16x + 64 = 0 \) is:
\[
\boxed{8}
\]
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