Questions: x^6-26x^3-27

Transcript text: $x^{6}-26 x^{3}-27$

Solution

Solution Steps

To solve the polynomial equation $x^{6} - 26x^{3} - 27 = 0$, we can use a substitution method. Let $y = x^3$, which transforms the equation into a quadratic form: $y^2 - 26y - 27 = 0$. We can then solve this quadratic equation for $y$ using the quadratic formula. Once we find the values of $y$, we substitute back to find the corresponding values of $x$.

Step 1: Transform the Original Equation

The original equation is $x^6 - 26x^3 - 27 = 0$. We use the substitution $y = x^3$ to transform it into a quadratic equation: \[ y^2 - 26y - 27 = 0 \]

Step 2: Solve the Quadratic Equation

To solve the quadratic equation $y^2 - 26y - 27 = 0$, we use the quadratic formula: \[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where $a = 1$, $b = -26$, and $c = -27$.

The discriminant is: \[ b^2 - 4ac = (-26)^2 - 4 \times 1 \times (-27) = 676 + 108 = 784 \]

The solutions for $y$ are: \[ y_1 = \frac{26 + \sqrt{784}}{2} = \frac{26 + 28}{2} = 27 \] \[ y_2 = \frac{26 - \sqrt{784}}{2} = \frac{26 - 28}{2} = -1 \]

Step 3: Solve for $x$

Since $y = x^3$, we solve for $x$ by taking the cube root of each solution for $y$.

For $y_1 = 27$: \[ x_1 = \sqrt[3]{27} = 3 \]

For $y_2 = -1$: \[ x_2 = \sqrt[3]{-1} = -1 \]