Solving Trinomials: Third power

To solve the problem $3x^3 - 10x^2 + 7x = 0$, follow these steps:

• Step 1: Factor out the greatest common factor (GCF). The common factor in all terms is $x$. Factoring out $x$ gives:

$x(3x^2 - 10x + 7) = 0$

• Step 2: Apply the Zero Product Property. Set each factor equal to zero:

$x = 0$ or $3x^2 - 10x + 7 = 0$

• Step 3: Solve the quadratic equation $3x^2 - 10x + 7 = 0$ using the quadratic formula:

The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Here, $a = 3$, $b = -10$, $c = 7$.

Calculate the discriminant:

$b^2 - 4ac = (-10)^2 - 4(3)(7) = 100 - 84 = 16$

Since the discriminant is a perfect square, this quadratic has rational roots. Using the quadratic formula gives:

$x = \frac{-(-10) \pm \sqrt{16}}{6} = \frac{10 \pm 4}{6}$

Thus, the solutions are:

$x = \frac{10 + 4}{6} = \frac{14}{6} = \frac{7}{3}$

$x = \frac{10 - 4}{6} = \frac{6}{6} = 1$

• Step 4: Combine all solutions. The solutions to the original equation are:

$x = 0$, $x = 1$, and $x = \frac{7}{3}$.

Therefore, the solution to the problem is $x = 0, 1, \frac{7}{3}$.

To solve the equation $x^3 + x^2 - 12x = 0$, follow these steps:

• Step 1: Factor out the greatest common factor. The common factor here is $x$.
• Step 2: The equation becomes $x(x^2 + x - 12) = 0$.
• Step 3: Apply the zero-product property. This gives us two equations to solve: $x = 0$ and $x^2 + x - 12 = 0$.
• Step 4: Solve $x = 0$. This is a straightforward solution: $x = 0$.
• Step 5: Solve the quadratic equation $x^2 + x - 12 = 0$. We will factor it:
• Factor as $(x - 3)(x + 4) = 0$.
• Set each factor equal to zero: $x - 3 = 0$ or $x + 4 = 0$.
• Solving these, we obtain $x = 3$ and $x = -4$.

Therefore, the solutions to the equation are $x = 0$, $x = 3$, and $x = -4$.

Thus, the complete solution set for $x$ is $x = 0, 3, -4$.

To solve the given cubic equation $x^3 - 7x^2 + 6x = 0$, follow these steps:

• Step 1: Identify that the equation can be factored by its Greatest Common Factor (GCF).

There is an $x$ common in all terms: $x(x^2 - 7x + 6) = 0$

• Step 2: Factor the quadratic expression $x^2 - 7x + 6$.

Look for two numbers that multiply to $6$ (the constant term) and add up to $-7$ (the coefficient of the linear term). The numbers are $-1$ and $-6$. Thus:

$x^2 - 7x + 6 = (x - 1)(x - 6)$

• Step 3: Set each factor equal to zero to solve for $x$.

Now that the equation is fully factored as $x(x - 1)(x - 6) = 0$, apply the zero product property:

$x = 0$, $x - 1 = 0$ (so $x = 1$), $x - 6 = 0$ (so $x = 6$)

Thus, the solutions to the equation $x^3 - 7x^2 + 6x = 0$ are $x = 0$, $x = 1$, and $x = 6$.