Solve by completing the square. $\newline$ $v^2 + 6v - 41 = 0$ $\newline$ Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth. $\newline$ $v =$ _ or $v =$ _

View step-by-step help

Home

Math Problems

Algebra 2

Solve a quadratic equation by completing the square

Full solution

Q. Solve by completing the square.

\newline

v^2 + 6v - 41 = 0

\newline

Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.

\newline

v =

_____ or

v =

_____

Rewrite in standard form: $v^2 + 6v - 41 = 0$ $\newline$ Rewrite the equation in the form of $x^2 + bx = c$ . $\newline$ Add $41$ to both sides to set the equation up for completing the square. $\newline$ $v^2 + 6v - 41 + 41 = 0 + 41$ $\newline$ $v^2 + 6v = 41$
Complete the square: $v^2 + 6v = 41$ $\newline$ Choose the equation after completing the square. $\newline$ Since $(6/2)^2 = 9$ , add $9$ to both sides to complete the square. $\newline$ $v^2 + 6v + 9 = 41 + 9$ $\newline$ $v^2 + 6v + 9 = 50$
Factor left side: $v^2 + 6v + 9 = 50$ $\newline$ Identify the equation after factoring the left side. $\newline$ $v^2 + 6v + 9 = 50$ $\newline$ $(v + 3)^2 = 50$
Take square root: $v + 3)^2 = 50(\newline$ Identify the equation after taking the square root on both sides. $\newline$ Take the square root of both sides of the equation. $\newline$ \$\sqrt{(v + 3)^2} = \sqrt{50}\)$\newline$$v + 3 = \pm\sqrt{50}$$\newline$$v + 3 \approx \pm 7.07$ (rounded to the nearest hundredth)
Isolate variable: We found:$\newline$$v + 3 \approx \pm 7.07$$\newline$Choose the equation after isolating the variable $v$.$\newline$To isolate $v$, subtract $3$ from both sides of the equation.$\newline$$v + 3 - 3 \approx \pm 7.07 - 3$$\newline$$v \approx -3 \pm 7.07$
Isolate variable: We found:$\newline$$v + 3 \approx \pm 7.07$$\newline$Choose the equation after isolating the variable $v$.$\newline$To isolate $v$, subtract $3$ from both sides of the equation.$\newline$$v + 3 - 3 \approx \pm 7.07 - 3$$\newline$$v \approx -3 \pm 7.07$We have:$\newline$$v \approx -3 \pm 7.07$$\newline$What are the two values of $v$?$\newline$$v \approx -3 + 7.07$ implies $v \approx 4.07$.$\newline$$v$$0$ implies $v$$1$.$\newline$Values of $v$: $v$$3$, $v$$4$