Question

now, write $x^2 - 16x + 64 = -8$ as
$(x - 8)^2 = -16$
$(x - 8)^2 = -8$
$(x - 16)^2 = -16$
$(x - 16)^2 = -8$

Explanation:

Step1: Recall perfect square formula

The perfect square trinomial formula is \((a - b)^2 = a^2 - 2ab + b^2\). For the quadratic expression \(x^2-16x + 64\), we compare it with \(a^2-2ab + b^2\). Here, \(a=x\), and \(-2ab=-16x\). Substituting \(a = x\) into \(-2ab=-16x\), we get \(-2\times x\times b=-16x\), which simplifies to \(b = 8\) (dividing both sides by \(- 2x\), \(x
eq0\)). Then \(b^2=8^2 = 64\), so \(x^2-16x + 64=(x - 8)^2\).

Step2: Substitute into the equation

The original equation is \(x^2-16x + 64=-8\). Since \(x^2-16x + 64=(x - 8)^2\), we substitute this into the equation, getting \((x - 8)^2=-8\)? Wait, no, wait. Wait, the right - hand side of the original equation is \(-8\)? Wait, no, let's check again. Wait, the original equation is \(x^2-16x + 64=-8\)? Wait, no, maybe I made a mistake. Wait, the quadratic expression \(x^2-16x + 64\) is a perfect square, \((x - 8)^2=x^2-16x + 64\). So the equation \(x^2-16x + 64=-8\) becomes \((x - 8)^2=-8\)? Wait, but let's check the options. Wait, the first option is \((x - 8)^2=-16\), the second is \((x - 8)^2=-8\), the third is \((x - 16)^2=-16\), the fourth is \((x - 16)^2=-8\). Wait, maybe there is a typo in my initial thought. Wait, let's re - evaluate the original equation. Wait, if the equation is \(x^2-16x+64=-8\), then \((x - 8)^2=-8\). But let's check the perfect square again. \((x - 8)^2=x^2-16x + 64\), so substituting into \(x^2-16x + 64=-8\) gives \((x - 8)^2=-8\). But wait, maybe the original equation was \(x^2-16x+64=-16\)? No, the problem says \(x^2 - 16x+64=-8\). Wait, but according to the perfect square, \(x^2-16x + 64=(x - 8)^2\), so the equation \(x^2-16x + 64=-8\) is equivalent to \((x - 8)^2=-8\). But let's check the options. The second option is \((x - 8)^2=-8\). Wait, but maybe I misread the original equation. Wait, the user's problem shows the equation \(x^2-16x + 64=-8\). So using the perfect square formula \((a - b)^2=a^2-2ab + b^2\), with \(a = x\), \(2ab = 16x\), so \(b = 8\), so \(x^2-16x + 64=(x - 8)^2\), and the equation is \((x - 8)^2=-8\).

Answer:

\((x - 8)^2=-8\) (corresponding to the second option: \((x - 8)^2=-8\))