Answered

Westonci.ca offers quick and accurate answers to your questions. Join our community and get the insights you need today. Get immediate and reliable solutions to your questions from a community of experienced experts on our Q&A platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.

Fill in the missing reason in the proof of the quadratic formula.

\begin{tabular}{|l|l|}
\hline \multicolumn{1}{|c|}{Statements} & \multicolumn{1}{|c|}{Reasons} \\
\hline
[tex]$a x^2 + b x + c = 0$[/tex] & Given \\
\hline
[tex]$a x^2 + b x = -c$[/tex] & Subtract [tex]$c$[/tex] from both sides of the equation \\
\hline
[tex]$x^2 + \frac{b}{a} x = -\frac{c}{a}$[/tex] & Divide both sides of the equation by [tex]$a$[/tex] \\
\hline
[tex]$x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2$[/tex] & Complete the square and add [tex]$\left(\frac{b}{2a}\right)^2$[/tex] to both sides \\
\hline
[tex]$x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2}$[/tex] & Square [tex]$\left(\frac{b}{2a}\right)$[/tex] on the right side of the equation \\
\hline
[tex]$x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = -\frac{4ac}{4a^2} + \frac{b^2}{4a^2}$[/tex] & Find a common denominator on the right side of the equation \\
\hline
[tex]$x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$[/tex] & Add the fractions together on the right side of the equation \\
\hline
[tex]$\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$[/tex] & Factor the left side of the equation \\
\hline
\end{tabular}

Sagot :

GinnyAnswer
Certainly! Let's walk through the process step-by-step in detail.

We start with the general quadratic equation:

1. Statement: [tex]\( ax^2 + bx + c = 0 \)[/tex]
Reason: Given

2. Statement: [tex]\( ax^2 + bx = -c \)[/tex]
Reason: Subtract [tex]\( c \)[/tex] from both sides of the equation

3. Statement: [tex]\( x^2 + \frac{b}{a}x = -\frac{c}{a} \)[/tex]
Reason: Divide both sides of the equation by [tex]\( a \)[/tex]

4. Statement: [tex]\( x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2 \)[/tex]
Reason: Complete the square by adding [tex]\(\left(\frac{b}{2a}\right)^2\)[/tex] to both sides

5. Statement: [tex]\( x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2} \)[/tex]
Reason: Square [tex]\(\left(\frac{b}{2a}\right)^2\)[/tex] on the right side of the equation

6. Statement: [tex]\( x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = -\frac{4ac}{4a^2} + \frac{b^2}{4a^2} \)[/tex]
Reason: Find a common denominator on the right side of the equation

7. Statement: [tex]\( x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} \)[/tex]
Reason: Add the fractions together on the right side of the equation

8. Statement: [tex]\( \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} \)[/tex]
Reason: Rewrite the left side as a square of the binomial

By following these steps, we transform the quadratic equation into a form that allows us to recognize the left side as a perfect square trinomial, making it easier to further solve the equation using the quadratic formula.
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.