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Chapter 11: Problem 50

Solve. Use a calculator to approximate, to three decimal places, the solutions as rational numbers. $$ 3 x^{2}-3 x-2=0 $$

Short Answer

Expert verified
\[x \approx 1.550\] and \[x \approx -0.217\].

Step by step solution

01

- Identify the quadratic equation

The given equation is a quadratic equation in the form of \[ax^2 + bx + c = 0\] where \[a = 3\], \[b = -3\], and \[c = -2\].
02

- Use the quadratic formula

The quadratic formula is \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. Substitute the values of \[a\], \[b\], and \[c\] into the formula.
03

- Calculate the discriminant

Substitute \[a = 3\], \[b = -3\], and \[c = -2\] into the discriminant formula \[b^2 - 4ac\]:\[(-3)^2 - 4(3)(-2)\].Simplify to: \[9 + 24 = 33\].
04

- Simplify the quadratic formula

Using the discriminant \[33\], proceed with the quadratic formula:\[x = \frac{-(-3) \pm \sqrt{33}}{2(3)}\] simplifies to \[x = \frac{3 \pm \sqrt{33}}{6}\].
05

- Calculate the roots

Calculate the approximate solutions using a calculator:\[x_1 = \frac{3 + \sqrt{33}}{6} \approx 1.550\]\[x_2 = \frac{3 - \sqrt{33}}{6} \approx -0.217\].

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Key Concepts

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

quadratic formula
The quadratic formula is a powerful tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \).
It helps find the values of \( x \) that satisfy the equation. The formula is expressed as follows:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here鈥檚 what each part means:
  • \textbf{\( a \):} The coefficient of \( x^2 \)
  • \textbf{\( b \):} The coefficient of \( x \)
  • \textbf{\( c \):} The constant term
The quadratic formula can handle any quadratic equation, whether it has real roots, complex roots, or a single repeated root.
Make sure to substitute the values of \( a, b, \) and \( c \) carefully to find accurate solutions.
discriminant
The discriminant is a specific part of the quadratic formula. It鈥檚 the expression inside the square root: \( b^2 - 4ac \).
The discriminant reveals important information about the nature of the roots:
  • If the discriminant is positive (greater than 0), the quadratic equation has two distinct real roots.
  • If the discriminant is zero, the quadratic equation has exactly one real root (a repeated root).
  • If the discriminant is negative (less than 0), the quadratic equation has two complex roots.
For the given exercise:
  • \textbf{\( a \) = 3}
  • \textbf{\( b \) = -3}
  • \textbf{\( c \) = -2}
Substituting these into the discriminant formula, we get:
\[ (-3)^2 - 4(3)(-2) = 9 + 24 = 33 \]
Since 33 is positive, the equation \( 3x^2 - 3x - 2 = 0 \) has two distinct real roots.
approximating roots
After finding the discriminant and using the quadratic formula, we arrive at the expression that gives the roots:
\[ x = \frac{3 \pm \sqrt{33}}{6} \]
To get the exact values of the roots, we need to handle the square root and the arithmetic.
This is often done with a calculator to get precise decimal values.
The approximate roots are calculated as follows:
  • \textbf{First Root (\( x_1 \)):} Simplify \( \frac{3 + \sqrt{33}}{6} \). Using a calculator for the square root of 33, we find:
    \[ x_1 \approx \frac{3 + 5.745}{6} \approx \frac{8.745}{6} \approx 1.550 \]
  • \textbf{Second Root (\( x_2 \)):} Simplify \( \frac{3 - \sqrt{33}}{6} \). Using the same square root value, we find:
    \[ x_2 \approx \frac{3 - 5.745}{6} \approx \frac{-2.745}{6} \approx -0.217 \]
Therefore, the solutions to the quadratic equation \( 3x^2 - 3x - 2 = 0 \) are approximately \( x_1 \approx 1.550 \) and \( x_2 \approx -0.217 \).

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Most popular questions from this chapter

Solve. $$ 9 x^{2}-16=0 $$

Write an equation for a function having a graph with the same shape as the graph of \(f(x)=\frac{3}{5} x^{2},\) but with the given point as the vertex. $$ (-4,-2) $$

Solve by completing the square. Show your work. $$ t^{2}+6 t-5=0 $$

If \(f(x)=\left(x-\frac{1}{3}\right)\left(x^{2}+6\right)\) and \(g(x)=\) \(\left(x-\frac{1}{3}\right)\left(x^{2}-\frac{2}{3}\right),\) find all \(a\) for which \((f+g)(a)=0\).

If the graphs of two quadratic functions have the same \(x\) -intercepts, will they also have the same vertex? Why or why not?
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