/*! 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 14 Solve. $$ t^{2}+4 t=1 $$... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 14

Solve. $$ t^{2}+4 t=1 $$

Short Answer

Expert verified
The solutions are \( t = -2 + \, \text{鈭�5} \) and \( t = -2 - \, \text{鈭�5} \).

Step by step solution

01

Move the Constant Term to the Left Side

Begin by re-arranging the equation to set it to zero. Subtract 1 from both sides:\[ t^2 + 4t - 1 = 0 \]
02

Apply the Quadratic Formula

To solve for \(t\), use the quadratic formula \( t = \frac{{-b \, \text{卤} \, \sqrt{{b^2 - 4ac}}}}{{2a}} \). Here, \(a = 1\), \(b = 4\), and \(c = -1\).
03

Calculate the Discriminant

First, calculate the discriminant, \( \, \text{螖} = b^2 - 4ac \):\[ \, \text{螖} = 4^2 - 4 \times 1 \times (-1) = 16 + 4 = 20 \]
04

Find the Square Root of the Discriminant

Compute the square root of the discriminant:\[ \, \text{鈭�20} = 2 \, \text{鈭�5} \]
05

Substitute Back into the Quadratic Formula

Substitute \(b = 4\), \( \, \text{鈭�20} = 2 \, \text{鈭�5} \) back into the quadratic formula:\[ t = \frac{{-4 \, \text{卤} \, 2 \, \text{鈭�5}}}{2} \]
06

Simplify the Expression

Simplify the fraction:\[ t = -2 \, \text{卤} \, \text{鈭�5} \]

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Key Concepts

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

quadratic formula
To solve any quadratic equation of the form \(ax^2 + bx + c = 0\), we can use the quadratic formula. This formula is given by:

\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula helps us find the roots (solutions) of the quadratic equation.

Here鈥檚 a breakdown of the parts:
  • a, b, and c: The coefficients of the equation where \(a\) is the coefficient of \(x^2\), \(b\) is the coefficient of \(x\), and \(c\) is the constant term.
  • Discriminant (\(\Delta\)): This part is inside the square root: \(b^2 - 4ac\). It tells us about the nature of the roots.
  • \( \pm \) symbol: This means there are typically two solutions, one with addition and one with subtraction.
discriminant
The discriminant is a key part of the quadratic formula. It is given by \(b^2 - 4ac\) and is denoted by the symbol \( \Delta\).

The value of the discriminant determines the nature of the roots:
  • If \( \Delta > 0\): There are two real and distinct roots.
  • If \( \Delta = 0\): There is exactly one real root (also called a repeated or double root).
  • If \( \Delta < 0\): There are two complex roots (they are not real numbers).

In the given exercise, the discriminant \( \Delta = 20\) was calculated using: \( - 4ac = 16 + 4 = 20\). Since \( \Delta > 0\), it indicates two real and distinct solutions.
simplification steps
Simplification steps are final adjustments to make sure the equation is easily interpreted. This includes:
  • Combining like terms
  • Reducing fractions
  • Making sure the solutions are in their simplest form

In the exercise, the simplified form of the quadratic equation solution was: \( t = -2 \pm \sqrt{5}\)

Here鈥檚 a more detailed breakdown of the simplification steps:
After substituting the discriminant back into the quadratic formula:

\( t = \frac{-4 \pm 2\sqrt{5}}{2} \)

We then simplify the fraction by dividing the numerator terms by the denominator:

\( t = -2 \pm \sqrt{5}\))

This final step ensures we have the simplest form of the solutions.

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Most popular questions from this chapter

Complete the square to find the \(x\) -intercepts of each function given by the equation listed. $$ g(x)=x^{2}+5 x+2 $$

To prepare for Section \(11.2,\) review evaluating expressions and simplifying radical expressions (Sections \(1.8,10.3\) and \(10.8)\) Simplify. \([10.3],[10.8]\) $$ \sqrt{-4} $$

Solve. $$ t^{2}=144 $$

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. $$ (4,1) $$

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