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Chapter 11: Problem 15

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

Short Answer

Expert verified
\( x = \frac{3 + \sqrt{29}}{2} \) and \( x = \frac{3 - \sqrt{29}}{2} \).

Step by step solution

01

- Move all terms to one side

To solve the equation, start by moving all terms to one side to set the equation to zero: \[ x^{2} - 3x - 5 = 0 \]
02

- Identify coefficients

The equation is now in the standard quadratic form \( ax^2 + bx + c = 0 \). Identify the coefficients: \( a = 1 \), \( b = -3 \), \( c = -5 \).
03

- Apply the quadratic formula

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve for \( x \). Substituting the values, we get: \[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-5)}}{2(1)} \] Simplify inside the square root: \[ x = \frac{3 \pm \sqrt{9 + 20}}{2} \] This simplifies further to: \[ x = \frac{3 \pm \sqrt{29}}{2} \]
04

- Write the solutions

The solutions to the quadratic equation are: \[ x = \frac{3 + \sqrt{29}}{2} \] and \[ x = \frac{3 - \sqrt{29}}{2} \]

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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. It provides a way to find the roots of any quadratic equation, which is an equation of the form \[ ax^2 + bx + c = 0 \]The formula itself is stated as \[ x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a} \] Where:
  • \( a \): The coefficient of the quadratic term \( x^2 \)
  • \( b \): The coefficient of the linear term \( x \)
  • \( c \): The constant term
When solving a quadratic equation, you simply need to substitute the values of \( a \), \( b \), and \( c \) into this formula and solve. This helps you find the values of \( x \) that satisfy the equation.
Standard Quadratic Form
Before applying the quadratic formula, the equation must be in standard quadratic form. This form looks like this: \[ ax^2 + bx + c = 0 \] In this form:
  • The term \( ax^2 \) is the quadratic term.
  • The term \( bx \) is the linear term.
  • The term \( c \) is the constant.
To convert an equation to this form, you may need to move all terms to one side of the equation so that the other side equals zero. For example, the equation: \[ x^2 = 3x + 5 \] Can be rewritten as: \[ x^2 - 3x - 5 = 0 \] This is now in the standard quadratic form, and you can proceed to identify the coefficients \( a \), \( b \), and \( c \).
Identifying Coefficients
Identifying the coefficients in the standard quadratic form is a crucial step. Once the equation is in the form \[ ax^2 + bx + c = 0 \] You can easily identify the values of \( a \), \( b \), and \( c \). In the example equation \[ x^2 - 3x - 5 = 0 \] The coefficients are:
  • \( a = 1 \)
  • \( b = -3 \)
  • \( c = -5 \)
These coefficients are then used in the quadratic formula. Finding these coefficients correctly is very important because any mistake in identification will lead to incorrect solutions for the quadratic equation. Make sure to carefully review the equation and ensure it is properly in standard form before identifying the coefficients.

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