Question

solve for x.
-7x = 3x² - 6
if there is more than one solution, separate them with commas.
if there is no solution, click on
o solution\.
x =
(there is a box for input and a calculator - like interface with buttons for fractions, mixed numbers, no solution, etc.)

Explanation:

Step1: Rearrange the equation

First, we rearrange the given equation \(-7x = 3x^2 - 6\) into standard quadratic form \(ax^2 + bx + c = 0\).
Subtract \(-7x\) from both sides (or add \(7x\) to both sides) to get:
\(3x^2 + 7x - 6 = 0\)
Here, \(a = 3\), \(b = 7\), and \(c = -6\).

Step2: Use the quadratic formula

The quadratic formula is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).
First, calculate the discriminant \(\Delta=b^{2}-4ac\).
Substitute \(a = 3\), \(b = 7\), and \(c = -6\) into the discriminant formula:
\(\Delta=(7)^{2}-4\times3\times(-6)\)
\(= 49 + 72\)
\(= 121\)

Then, substitute \(a = 3\), \(b = 7\), and \(\Delta = 121\) into the quadratic formula:
\(x=\frac{-7\pm\sqrt{121}}{2\times3}\)
Since \(\sqrt{121}=11\), we have:
\(x=\frac{-7\pm11}{6}\)

Step3: Find the two solutions

For the plus sign:
\(x=\frac{-7 + 11}{6}=\frac{4}{6}=\frac{2}{3}\)
For the minus sign:
\(x=\frac{-7-11}{6}=\frac{-18}{6}=-3\)

Answer:

\(\frac{2}{3}, -3\)