Question

select the correct answer. use a graphing tool to solve the equation for x. $2^{x}-4=-4^{x}+4$ a. $x\approx1.25$ b. $x\approx2.25$ c. $x\approx - 1.5$ d. $x\approx0.25$

Explanation:

Step1: Rearrange the equation

$2^x + 4^x = 4 + 4$
$2^x + (2^2)^x = 8$
$2^x + (2^x)^2 = 8$

Step2: Substitute $y=2^x$

$y^2 + y - 8 = 0$

Step3: Solve quadratic equation

Usando la fórmula cuadrática $y=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ para $ay^2+by+c=0$:
$y=\frac{-1\pm\sqrt{1^2-4(1)(-8)}}{2(1)}=\frac{-1\pm\sqrt{33}}{2}$

Step4: Take positive root (since $2^x>0$)

$y=\frac{-1+\sqrt{33}}{2}\approx\frac{-1+5.7446}{2}\approx2.3723$

Step5: Solve for $x$

$x=\log_2(y)=\frac{\ln(y)}{\ln(2)}\approx\frac{\ln(2.3723)}{\ln(2)}\approx\frac{0.864}{0.693}\approx1.25$

Answer:

A. $x \approx 1.25$