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Chapter 1: Problem 15

Find the zeros of the function algebraically. $$f(x)=2 x^{2}-7 x-30$$

Short Answer

Expert verified
The zeroes of the function \(f(x)\) are \(x = \frac{7+\sqrt{289}}{4}\) and \(x = \frac{7-\sqrt{289}}{4}\)

Step by step solution

01

Setting Up Equation

First, set the given quadratic equation \(2x^{2} - 7x - 30 = 0\) since we are finding the zeroes. We also note that \(a = 2\), \(b = -7\) and \(c = -30 \) for later use in the quadratic formula if necessary.
02

Factoring

Attempt to factor this quadratic expression to easily find its roots. However, in this case, it is seen that the equation does not factorize comfortably to integers. Prime factorization of the constant term 30 does not help to find a pair that adds up to -7 as the coefficient of x. Therefore, it is easier to resort to the usage of the quadratic formula.
03

Solve Using Quadratic Formula

Plug the values of a, b and c into the quadratic formula:\(x = \frac{-(-7) \pm \sqrt{(-7)^{2}-4*(2)*(-30)}}{2*(2)}\)Simplify this to obtain:\(x = \frac{7 \pm \sqrt{49+240}}{4}\)Further simplification gives us the roots, or zeroes, of the function.

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Most popular questions from this chapter

A rectangle is bounded by the \(x\) -axis and the semicircle \(y=\sqrt{36-x^{2}}\) (see figure). Write the area \(A\) of the rectangle as a function of \(x,\) and graphically determine the domain of the function.

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