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Chapter 2: Problem 74

Use the Quadratic Formula to solve the quadratic equation. $$9 x^{2}-6 x+37=0$$

Short Answer

Expert verified
The solutions for the equation \(9x^2 - 6x + 37 = 0\) are \(x = 1/3 + 2i\) and \(x = 1/3 - 2i\).

Step by step solution

01

Identify the Coefficients

In the equation \(9x^2 - 6x + 37 = 0\), the coefficients are \(a = 9\), \(b = -6\), and \(c = 37\). These are the numbers that replace \(a\), \(b\), and \(c\) in the quadratic formula.
02

Calculate the Discriminant

The discriminant is part of the quadratic formula under the square root, given by \(b^2 - 4ac\). In this case it equals \((-6)^2 - 4 * 9 * 37\), which simplifies to \(36 - 1332 = -1296\).
03

Apply the Quadratic Formula

Substitute \(a\), \(b\), and the discriminant into the quadratic formula, given by \(x = [-b \pm \sqrt{discriminant}] / (2a)\). The result will return two solutions, \(x1\) and \(x2\), given by \(x = [6 \pm \sqrt{-1296}]/(2 * 9)\), which simplifies to \(x = [6 \pm 36i]/18\), or \(x = 1/3 \pm 2i\). Since the discriminant is less than zero, the solutions are complex numbers.

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Most popular questions from this chapter

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$f(x)=16 x^{3}-20 x^{2}-4 x+15$$

Prove that the complex conjugate of the sum of two complex numbers \(a_{1}+b_{1} i\) and \(a_{2}+b_{2} i\) is the sum of their complex conjugates.

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$h(x)=2 x^{3}+3 x^{2}+1$$

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