Question

use a graphing calculator to solve the equation by graphing. if necessary, round to the nearest hundredth. list solutions from least to greatest. $x^{8} = -x^{7} + 3$

Explanation:

Step1: Rearrange the equation

Rewrite the equation \(x^{8} = -x^{7}+3\) as \(x^{8}+x^{7}-3 = 0\). We need to find the roots of the function \(y = x^{8}+x^{7}-3\) by graphing.

Step2: Analyze the function's behavior

• For large positive \(x\), \(x^{8}\) dominates, so \(y\) approaches \(+\infty\).
• For large negative \(x\), \(x^{8}\) (even power) is positive and large, \(x^{7}\) (odd power) is negative and large in magnitude, but \(x^{8}\) dominates, so \(y\) approaches \(+\infty\).
• At \(x = 0\), \(y=0 + 0-3=-3\).
• At \(x = 1\), \(y=1 + 1-3=-1\).
• At \(x = 2\), \(y = 256+128 - 3=381\).
• At \(x=- 1\), \(y = 1-1 - 3=-3\).
• At \(x=-2\), \(y = 256-128 - 3 = 125\).

Step3: Graph the function (conceptually)

We can use a graphing calculator to plot \(y=x^{8}+x^{7}-3\). The roots are the \(x\)-intercepts. By using a graphing calculator (or software like Desmos), we find that the function crosses the \(x\)-axis at approximately \(x\approx - 1.46\) and \(x\approx1.14\) (after checking the graph and using the zero - finding feature of the calculator).

Answer:

\(-1.46\), \(1.14\) (rounded to the nearest hundredth)