/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 17 Solve: \(3 x^{2}-4 x-5=0\)... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 17

Solve: \(3 x^{2}-4 x-5=0\)

Short Answer

Expert verified
The solutions are \(x = \frac{2 + \sqrt{19}}{3}\) and \(x = \frac{2 - \sqrt{19}}{3}\).

Step by step solution

01

Identify coefficients

The quadratic equation is given by \(3x^2 - 4x - 5 = 0\). Identify the coefficients: \(a = 3\), \(b = -4\), \(c = -5\).
02

Use the quadratic formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Plug in the values of \(a\), \(b\), and \(c\). \(x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(3)(-5)}}{2(3)}\).
03

Simplify inside the square root

Calculate inside the square root: \((-4)^2 = 16\) and \(4ac = 4(3)(-5) = -60\). So, the discriminant becomes \(16 - (-60) = 16 + 60 = 76\). Now, plug this back into the formula: \(x = \frac{4 \pm \sqrt{76}}{6}\).
04

Simplify the expression

\(76\) can be expressed as \(4 \times 19\), so \( \sqrt{76} = \sqrt{4 \times 19} = 2 \sqrt{19} \). Substitute this back into the formula: \(x = \frac{4 \pm 2\sqrt{19}}{6}\). Simplify by dividing both terms in the numerator by 2: \(x = \frac{2 \pm \sqrt{19}}{3}\).
05

Write the final solutions

The solutions are: \(x = \frac{2 + \sqrt{19}}{3}\) and \(x = \frac{2 - \sqrt{19}}{3}\).

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Key Concepts

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

Coefficients
In a quadratic equation of the form ax^2 + bx + c = 0, the values of a, b, and c are known as the coefficients. These coefficients dictate the parabola's shape and position on a graph.

To identify these, simply match the quadratic equation you鈥檙e given to this standard form. For example, with the equation 3x^2 - 4x - 5 = 0, the coefficients are:
  • a = 3,
  • b = -4,
  • c = -5.
Knowing these coefficients is crucial as they are used in the quadratic formula to find the solutions of the equation.
Quadratic Formula
One powerful tool for solving quadratic equations is the quadratic formula. This formula provides an exact solution to any quadratic equation, and it is given by:

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