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Chapter 0: Problem 11

Use the quadratic formula to solve the equations. $$\frac{3}{2} x^{2}-6 x+5=0$$

Short Answer

Expert verified
The solutions are \[x = 2 + \frac{\sqrt{6}}{3}\] and \[x = 2 - \frac{\sqrt{6}}{3}\].

Step by step solution

01

Identify coefficients

For the quadratic equation in the form of \[ax^2 + bx + c = 0\]the given equation \[\frac{3}{2}x^2 - 6x + 5 = 0\]has coefficients: a = \frac{3}{2}, b = -6, and c = 5.
02

Write down the quadratic formula

The quadratic formula is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
03

Calculate the discriminant

The discriminant is \[b^2 - 4ac\].Substitute a, b, and c: \[(-6)^2 - 4 \left( \frac{3}{2} \right) (5)\]:discriminant = 36 - 30 = 6.
04

Substitute into the quadratic formula

Substitute the values a, b, and the discriminant into the quadratic formula:\[x = \frac{-(-6) \pm \sqrt{6}}{2 \left( \frac{3}{2} \right)}\],which simplifies to:\[x = \frac{6 \pm \sqrt{6}}{3}\].
05

Solve for the roots

Simplify the expression:\[x = 2 \pm \frac{\sqrt{6}}{3}\].So the roots are:\[x = 2 + \frac{\sqrt{6}}{3}\] and \[x = 2 - \frac{\sqrt{6}}{3}\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coefficients
To solve a quadratic equation using the quadratic formula, we need to understand coefficients. In general, a quadratic equation looks like this: \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are the coefficients.
  • \(a\) is the coefficient of \(x^2\).
  • \(b\) is the coefficient of \(x\).
  • \(c\) is the constant term.
Identifying these coefficients from the equation is the first step. In the equation \(\frac{3}{2}x^2 - 6x + 5 = 0\), the coefficients are:
  • \(a = \frac{3}{2}\)
  • \(b = -6\)
  • \(c = 5\)
These coefficients are what we'll use in the quadratic formula next.
Discriminant
The discriminant helps us understand how many solutions a quadratic equation has. It is found inside the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), in the part under the square root: \(b^2 - 4ac\).
  • If the discriminant (\

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Most popular questions from this chapter

Use your graphing calculator to find the value of the given function at the indicated values of \(x .\) $$f(x)=x^{4}+2 x^{3}+x-5 ; \quad x=-\frac{1}{2}, x=3$$

Let \(f(x)=x^{6}, g(x)=\frac{x}{1-x},\) and \(h(x)=x^{3}-5 x^{2}+1 .\) Calculate the following functions. $$g(f(x))$$

Let \(f(x)=\sqrt[3]{x}\) and \(g(x)=\frac{1}{x^{2}} .\) Calculate the following functions. Take \(x > 0\). $$\frac{g(x)}{f(x)}$$

Let \(f(x)=x^{2} .\) Graph the functions \(f(x+1)\) \(f(x-1), f(x+2),\) and \(f(x-2) .\) Make a guess about the relationship between the graph of a general function \(f(x)\) and the graph of \(f(g(x)),\) where \(g(x)=x+a\) for some constant \(a\) Test your guess on the functions \(f(x)=x^{3}\) and \(f(x)=\sqrt{x}.\)

The expressions may be factored as shown. Find the missing factors. $$\sqrt{x}-\frac{1}{\sqrt{x}}=\frac{1}{\sqrt{x}}(\quad)$$
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