Question

Solve the following quadratic equation using the perfect square trinomial pattern: 64x2−80x+25=0
.
x=−0.625
x equals negative 0.625

x=25
x equals 25

x=0.625
x equals 0.625

x=−25

Answer 1

To solve the quadratic equation \(64x^2 - 80x + 25 = 0\) using the perfect square trinomial pattern, we first identify if it can be expressed in the form \((ax + b)^2 = 0\).

1. The given equation is \(64x^2 - 80x + 25\).
2. We can observe that the coefficient of \(x^2\) is \(64\), and therefore, we can factor it out.

We need to check if this quadratic can be expressed as a perfect square. The standard form for a perfect square trinomial is \((mx + n)^2\), where \(m^2\) is the coefficient of \(x^2\) and \(n^2\) is the constant term.

Taking the square roots:
- The square root of \(64\) is \(8\), so \(m = 8\).
- The constant term is \(25\) which is also a perfect square, \(5^2\), so \(n = 5\).

Next, we will check if \(2mn = -80\):

\[
2 \cdot 8 \cdot 5 = 80
\]

Since we want \(2mn\) to equal \(-80\), we actually have:

\[
2 \cdot (-8) \cdot 5 = -80
\]

Thus, we see it can be written as:

\[
(8x - 5)^2 = 0
\]

Now, we can solve the equation:

\[
(8x - 5)^2 = 0
\]

Taking the square root of both sides:

\[
8x - 5 = 0
\]

Now solving for \(x\):

\[
8x = 5 \quad \Rightarrow \quad x = \frac{5}{8} = 0.625
\]

Thus, the solution to the equation \(64x^2 - 80x + 25 = 0\) is:

\[
x = 0.625
\]

So, the correct option from the given choices is:

**x = 0.625**.