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Chapter 11: Problem 59

Explain how you could use the quadratic formula to help factor a quadratic polynomial.

Short Answer

Expert verified
Find the roots using the quadratic formula and then use them to write the polynomial in its factored form \(a(x - x_1)(x - x_2)\).

Step by step solution

01

Identify the Quadratic Equation

A quadratic polynomial is typically in the form of \(ax^2 + bx + c\). Write down the polynomial you want to factor.
02

Apply the Quadratic Formula

To find the roots of the quadratic equation \(ax^2 + bx + c = 0\), use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].
03

Calculate the Discriminant

Compute the discriminant \(\Delta = b^2 - 4ac\) to determine the nature of the roots.
04

Find the Roots

If the discriminant is non-negative, calculate the roots using \(x_1 = \frac{-b + \sqrt{\Delta}}{2a}\) and \(x_2 = \frac{-b - \sqrt{\Delta}}{2a}\).
05

Write the Factored Form

Using the roots \(x_1\) and \(x_2\), the factored form of the quadratic polynomial is \(a(x - x_1)(x - x_2)\).

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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 provides a way to find the roots of any quadratic equation. A quadratic equation is written as \(ax^2 + bx + c = 0\). The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. It takes the coefficients \(a\), \(b\), and \(c\) from your equation and solves for \(x\).
This formula is extremely useful when factoring quadratic polynomials because it directly gives you the polynomial's roots, or solutions.
These roots can then be used to express the quadratic in its factored form.
Remember to apply it carefully, following each step to avoid mistakes.
Discriminant
The discriminant \(\Delta\) is a crucial part of the quadratic formula. It is found within the square root, calculated as \(\Delta = b^2 - 4ac\). The discriminant tells us about the nature of the roots of the quadratic equation:
  • If \(\Delta > 0\), there are two distinct real roots.
  • If \(\Delta = 0\), there is exactly one real root (a repeated root).
  • If \(\Delta < 0\), there are no real roots; instead, there are two complex roots.

Calculating the discriminant is an essential step before proceeding with finding the roots, as it guides our expectations and next steps in the process.
Polynomial Roots
The roots (or solutions) of the polynomial are values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). These roots can be found using the quadratic formula. If the discriminant is non-negative, calculate the roots using: \[ x_1 = \frac{-b + \sqrt{\Delta}}{2a} \] and \[ x_2 = \frac{-b - \sqrt{\Delta}}{2a} \].
Once you have the roots, you know the values of \(x\) where the polynomial equals zero.
These roots are crucial because they allow you to re-write the polynomial in factored form, which is very useful for both solving and graphing the equation.
Factored Form
Once you have determined the roots from the quadratic formula, you can express the quadratic polynomial in its factored form.
If the roots are \(x_1\) and \(x_2\), the factored form of the polynomial \(ax^2 + bx + c\) is: \[ a(x - x_1)(x - x_2) \].
In this form, it's easy to see the roots and understand the behavior of the quadratic function.
This process is essential in both algebra and higher-level mathematics, as it simplifies working with polynomials and solving quadratic equations.

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Most popular questions from this chapter

To prepare for Section \(11.2,\) review evaluating expressions and simplifying radical expressions (Sections \(1.8,10.3\) and \(10.8)\) Evaluate. [ 1.8] $$ b^{2}-4 a c, \text { for } a=1, b=-1, \text { and } c=4 $$

Let \(F(x)=x^{2}+8 x+16 .\) Find \(x\) such that \(F(x)=9\).

Review finding intercepts and completing the square (Sections \(3.3,6.2,6.3, \text { and } 11.1)\) Find the \(x\) -intercepts. $$ y=x^{2}+8 x+15 $$

If the graphs of \(f(x)=a_{1}\left(x-h_{1}\right)^{2}+k_{1}\) and \(g(x)=a_{2}\left(x-h_{2}\right)^{2}+k_{2}\) have the same shape, what, if anything, can you conclude about the \(a^{\prime}\) s. the \(h^{\prime} s,\) and the \(k^{\prime} s ?\) Why?

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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