Questions: To solve (x^4-15 x^2+40=0), let (u=) Then rewrite the equation in terms of (u) as

Transcript text: To solve $x^{4}-15 x^{2}+40=0$, let $u=$ $\square$ Then rewrite the equation in terms of $u$ as $\square$

Solution

Solution Steps

To solve the equation $x^4 - 15x^2 + 40 = 0$, we can use a substitution method. Let $u = x^2$. This transforms the equation into a quadratic in terms of $u$.

Step 1: Substitution

To solve the equation $x^4 - 15x^2 + 40 = 0$, we use the substitution $u = x^2$. This transforms the equation into a quadratic equation in terms of $u$: \[ u^2 - 15u + 40 = 0 \]

Step 2: Solve the Quadratic Equation

We solve the quadratic equation $u^2 - 15u + 40 = 0$ using the quadratic formula: \[ u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where $a = 1$, $b = -15$, and $c = 40$.

Step 3: Calculate the Roots

Substituting the values into the quadratic formula, we find: \[ u = \frac{15 \pm \sqrt{15^2 - 4 \cdot 1 \cdot 40}}{2 \cdot 1} \] \[ u = \frac{15 \pm \sqrt{225 - 160}}{2} \] \[ u = \frac{15 \pm \sqrt{65}}{2} \]

Thus, the solutions for $u$ are: \[ u_1 = \frac{15 - \sqrt{65}}{2} \] \[ u_2 = \frac{15 + \sqrt{65}}{2} \]