/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 44 Solve the equation \(2 x^{3}-3 x... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 44

Solve the equation \(2 x^{3}-3 x^{2}-11 x+6=0\) given that \(-2\) is a zero of \(f(x)=2 x^{3}-3 x^{2}-11 x+6\)

Short Answer

Expert verified
The solutions to the equation are \(x = -2\), \(x = 1\), and \(x = 1.5\).

Step by step solution

01

Substitute known zero

As already given, -2 is a root of the equation. So, if we substitute -2 in place of x in the equation, it should equal zero. Doing this we have: \(2(-2)^{3}-3 (-2)^2-11(-2)+6 = 0\). As expected it returns zero, confirming that -2 is a root of the equation.
02

Factorize the equation

Knowing one root of the equation allows us to factorize it. We can divide the given expression by the root in order to get a quadratic equation. Dividing \(2 x^{3}-3 x^{2}-11 x+6\) by \((x+2)\) we get: \(2 x^{2} - 7x + 3\). Now, we have a simplified quadratic equation to solve.
03

Solve the quadratic equation

The quadratic equation \(2 x^{2} - 7x + 3 = 0\) can be solved using the Quadratic Formula \(-b \pm \sqrt{b^{2} - 4ac} / 2a\). Here a = 2, b = -7, c = 3. Upon substituting these values and solving, we get two roots of the equation. These are: \(x1 = 1\) and \(x2 = 1.5\).

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