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Chapter 5: Problem 54

Solve each equation. $$ 3 x^{2}+14 x=5 $$

Short Answer

Expert verified
x = \frac{1}{3} or x = -5

Step by step solution

01

Move all terms to one side

First, move the constant term to one side of the equation to have it in standard quadratic form. Subtract 5 from both sides:\[ 3 x^{2} + 14 x - 5 = 0 \]
02

Identify coefficients

Identify the coefficients for the quadratic formula. For the equation \(3 x^{2} + 14 x - 5 = 0\), the coefficients are: \(a = 3\), \(b = 14\), and \(c = -5\).
03

Quadratic Formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Plug in the values of \(a\), \(b\), and \(c\):\[ x = \frac{-14 \pm \sqrt{14^2 - 4(3)(-5)}}{2(3)} \]
04

Simplify inside the square root

Calculate the value inside the square root:\[ b^2 - 4ac = 14^2 - 4(3)(-5) = 196 + 60 = 256 \]So, the equation becomes:\[ x = \frac{-14 \pm \sqrt{256}}{6} \]
05

Simplify the square root

Simplify \( \sqrt{256} = 16 \) to get:\[ x = \frac{-14 \pm 16}{6} \]
06

Solve for the two possible values of x

Now, split it into two equations to find the two possible values of \( x \):\[ x = \frac{-14 + 16}{6} = \frac{2}{6} = \frac{1}{3} \]\[ x = \frac{-14 - 16}{6} = \frac{-30}{6} = -5 \]

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Key Concepts

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

Standard Quadratic Form
To solve quadratic equations effectively, we first need to understand the standard quadratic form. A quadratic equation is typically written as follows:

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