Find the interval in which the equation has at least one real root. $$x ^ { 4 } - 8 x ^ { 2 } + 15 = 0$$
A) $\quad - 1$ and 0
B) $- 4$ and $- 3$
C) 3 and 4
D) 2 and 3
E) $\quad 0$ and 1

Question

Quizplus · Accepted Answer

The given equation can be rewritten as a quadratic equation in terms of $y = x^2$, i.e., $y^2 - 8y + 15 = 0$. Factoring this equation gives $(y - 3)(y - 5) = 0$, which implies $y = 3$ or $y = 5$. Since $y = x^2$, and $x^2$ is always non-negative, the interval in which the original equation has at least one real root corresponds to the values of $x^2$ that are between 3 and 5, inclusive. Therefore, the correct interval is between 2 and $\sqrt{5}$, but since we are looking for integer intervals, the closest interval that contains at least one real root is between 2 and 3.

Find the Interval in Which the Equation Has at Least x4−8x2+15=0x ^ { 4 } - 8 x ^ { 2 } + 15 = 0x4−8x2+15=0

Find the Interval in Which the Equation Has at Least $x ^ { 4 } - 8 x ^ { 2 } + 15 = 0$