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Chapter 2: Problem 78

Use the Quadratic Formula to solve the quadratic equation. $$\frac{7}{8} x^{2}-\frac{3}{4} x+\frac{5}{16}=0$$

Short Answer

Expert verified
The solutions of the quadratic equation are \( x = \frac{ \frac{3}{4} + \frac{i}{16} }{ \frac{7}{4} }\) and \( x = \frac{ \frac{3}{4} - \frac{i}{16} }{ \frac{7}{4} }\).

Step by step solution

01

Identify a, b and c

Identify the values of a, b, and c in the quadratic equation. In this case, \(a = \frac{7}{8}\), \(b = -\frac{3}{4}\), and \(c = \frac{5}{16}\)
02

Substitute values into the formula

Substitute these values into the Quadratic Formula: \(-(-\frac{3}{4}) \pm \sqrt{(-\frac{3}{4})^{2}-4*\frac{7}{8}*\frac{5}{16}} \div 2*\frac{7}{8}\). After calculating, this becomes: \(\frac{3}{4} \pm \sqrt{\frac{9}{16}-\frac{35}{64}} \div \frac{7}{4}\)
03

Simplify the discriminant

The discriminant in the square root (\(\frac{9}{16}-\frac{35}{64}\)) simplifies to \(-\frac{1}{64}\). Since the discriminant is negative, it means our solutions will be complex.
04

Simplify the quadratic formula

Substitute \(-\frac{1}{64}\) back into the quadratic formula: \(\frac{3}{4} \pm \sqrt{-\frac{1}{64}} \div \frac{7}{4}\). This simplifies to \(\frac{3}{4} \pm \frac{i}{16} \div \frac{7}{4}\)
05

Solve for x

Lastly, divide \(\frac{3}{4} \pm \frac{i}{16}\) by \(\frac{7}{4}\) to get the final solutions.

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Most popular questions from this chapter

Find the rational zeros of the polynomial function. $$f(x)=x^{3}-\frac{1}{4} x^{2}-x+\frac{1}{4}=\frac{1}{4}\left(4 x^{3}-x^{2}-4 x+1\right)$$

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. $$f(x)=16 x^{3}-20 x^{2}-4 x+15$$

Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(x)=5 x^{3}-9 x^{2}+28 x+6$$

Find a polynomial function with real coefficients that has the given zeros. (There are many correct answers.) $$2,5+i$$

Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(y)=y^{4}-256$$
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