Question

Solve the equation. Tell which solution method you used. \(-14 x^{4}+118 x^{3}+72 x^{2}=0\)

Short answer

The solutions to the equation are \(x = 0\), \(x= \frac{-3}{7}\), and \(x=3\) using factoring and the quadratic formula.

Step-by-step solution

Facilitating the equation

Simplify the equation by factoring out the greatest common factor of all terms before any further calculations. The greatest common factor of \(-14x^{4}\), \(118x^{3}\), and \(72x^{2}\) is \(2x^{2}\). Factoring this out gives \[2x^{2}(-7x^{2} + 59x + 36) = 0\]

Setting each factor equal to zero

In order for the equation to be true, either \(2x^{2} = 0\) or \(-7x^{2} + 59x + 36=0\). These two can solve separately.

Solve the first factor

For the first factor \(2x^{2} = 0\). Simplify and solve the equation for x by dividing both sides by \(2\), which gives \(x = 0\).

Solve the second factor

For the second factor \(-7x^{2} + 59x + 36=0\) you can use any method of solving for quadratic equations such as factoring, completing the square, or the quadratic formula. In this case, using the quadratic formula which is given by \[x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\], where \(a=-7\), \(b=59\), and \(c=36\) is more convenient. Substituting these values into the formula and solving gives us \(x= \frac{-3}{7}, x=3\).