/*! 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 73 Solve by factoring. $$ x^{2}... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 73

Solve by factoring. $$ x^{2}-12 x+20=0 $$

Short Answer

Expert verified
The solutions are \(x = 2\) and \(x = 10\).

Step by step solution

01

Identify a Quadratic Equation

The given equation is in the form of a quadratic equation, which can be written as \( ax^2 + bx + c = 0 \). Here, \( a = 1 \), \( b = -12 \), and \( c = 20 \).
02

Set Up Factoring

To solve the equation by factoring, we need to find two numbers that multiply to \( c \) (which is 20) and add up to \( b \) (which is -12).
03

Find the Numbers

The numbers that multiply to 20 and add up to -12 are -2 and -10. This is because \((-2) \times (-10) = 20\) and \((-2) + (-10) = -12\).
04

Write the Factored Equation

Using the numbers from the previous step, we can rewrite the equation as \((x - 2)(x - 10) = 0\).
05

Solve for x

Set each factor equal to zero: 1. \(x - 2 = 0\) which gives \(x = 2\) 2. \(x - 10 = 0\) which gives \(x = 10\).
06

Verify the Solutions

To confirm, substitute \(x = 2\) and \(x = 10\) back into the original equation. Both values satisfy \(x^2 - 12x + 20 = 0\).

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Key Concepts

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

Factoring Method
The factoring method is a classic approach to solving quadratic equations. A quadratic equation is typically in the form \( ax^2 + bx + c = 0 \). The core idea of factoring involves expressing the quadratic expression as a product of two binomials. This allows you to find the solutions by solving simpler linear equations.

Here's how the factoring method works:
  • First, identify the coefficients \( a \), \( b \), and \( c \) from the quadratic equation.
  • Next, look for two numbers that multiply to \( c \) and add up to \( b \). These numbers are crucial because they help break the equation into simpler parts.
  • Rewrite the quadratic equation as a product of these two binomial expressions.
  • Finally, solve each binomial equation by setting it equal to zero and solving for \( x \). This will give you the roots of the equation.
When using this method, it's important to verify your solutions by substituting them back into the original equation to see if they satisfy it. Factoring is a powerful technique but may not work if the quadratic does not easily split into integers.
Quadratic Formula
The quadratic formula is a universal tool that provides a straightforward way to find the roots of any quadratic equation, even if the equation is difficult to factor. The general form of a quadratic equation is \( ax^2 + bx + c = 0 \), and the quadratic formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Here's a breakdown of how to use the quadratic formula:
  • First, ensure the equation is in standard form \( ax^2 + bx + c = 0 \).
  • Identify the values of \( a \), \( b \), and \( c \).
  • Calculate the discriminant \( b^2 - 4ac \). The discriminant tells us about the nature of the roots.
    • If it's positive, there are two real and distinct roots.
    • If it's zero, there is exactly one real root (a repeated root).
    • If it's negative, there are no real roots, but there are two complex roots.
  • Substitute \( a \), \( b \), and \( c \) into the quadratic formula to find the roots.
The quadratic formula is invaluable because it works for any quadratic equation, ensuring you can always find the roots without needing to factor.
Roots of Quadratic Equations
The roots of a quadratic equation are the values of \( x \) for which the equation \( ax^2 + bx + c = 0 \) has a value of zero. These roots are where the graph of the quadratic equation crosses the x-axis. Finding these roots is essential as they reveal crucial information about the equation's solutions.

There are three main methods to find the roots of quadratic equations:
  • Factoring: As mentioned, this involves rewriting the quadratic equation into a product of two binomials, then solving for \( x \).
  • Quadratic Formula: This is used when the equation is difficult to factor, ensuring that you can always solve the equation.
  • Completing the Square: A method where the equation is manipulated to form a perfect square trinomial, allowing you to solve for \( x \) directly.
The roots are fundamental to understanding the behavior of the quadratic function, helping in graphing and analyzing real-world scenarios modeled by quadratic equations. Always check your found roots by substituting them back into the original equation to ensure they solve it correctly.

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Most popular questions from this chapter

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