Question

Solve the equation. Tell which solution method you used. \(34 x^{4}-85 x^{3}+51 x^{2}=0\)

Short answer

The solutions for the equation \(34 x^{4}-85 x^{3}+51 x^{2}=0\) are \(x = 0, x = \frac{51}{34}, x = 1 \). The solution method used is factoring.

Step-by-step solution

Find out the Greatest common factor

The given equation is \(34 x^{4}-85 x^{3}+51 x^{2}=0\). First, find the greatest common factor of all the terms in the given equation. Here, \(x^{2}\) is the greatest common factor. Factor out \(x^{2}\) and rewrite the equation as: \(x^{2}(34x^{2} - 85x + 51) = 0\).

Factor the Quadratic Equation

Next, factor the quadratic equation \((34x^{2} - 85x + 51)\). We need to find two numbers whose product is \(34*51 = 1734\) and whose sum is \(-85\). After calculation we find that \(-34\) and \(-51\) are those two numbers. \(34x^{2} - 85x + 51 = 34x^{2} - 34x - 51x + 51 = 34x(x - 1) - 51(x - 1) = (34x - 51)(x - 1)\). So now we have \(x^{2}*(34x - 51)*(x - 1) = 0\).

Solve Each Equation for x

Now solve each of the factored equations for x. Set each factor equal to zero. Here we got three equations: \(x^{2} = 0, 34x - 51 = 0, x - 1 = 0\). Solving these equations will give us x = 0, x = \frac{51}{34}, x = 1.