Solve the following quadratic equation using the perfect square trinomial pattern: x^2 - 16x + 64 = 0

Answer 1

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:

$$\boxed{{\displaystyle 8}}$$