Question

question
use the quadratic formula to solve. express your answer in simplest form.
3y² - 10y - 10 = 3y
answer attempt 1 out of 3
additional solution no solution
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Explanation:

Step1: Rearrange the equation

First, we need to rewrite the given equation \(3y^{2}-10y - 10=3y\) in standard quadratic form \(ay^{2}+by + c = 0\). Subtract \(3y\) from both sides:
\(3y^{2}-10y-3y - 10=0\)
Simplify the like terms:
\(3y^{2}-13y - 10=0\)
Here, \(a = 3\), \(b=- 13\), \(c=-10\).

Step2: Apply the quadratic formula

The quadratic formula is \(y=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). Substitute \(a = 3\), \(b=-13\), \(c = - 10\) into the formula:
First, calculate the discriminant \(\Delta=b^{2}-4ac=(-13)^{2}-4\times3\times(-10)\)
\(\Delta = 169 + 120=289\)
Then, \(y=\frac{-(-13)\pm\sqrt{289}}{2\times3}=\frac{13\pm17}{6}\)

Step3: Find the two solutions

For the plus sign: \(y=\frac{13 + 17}{6}=\frac{30}{6}=5\)
For the minus sign: \(y=\frac{13-17}{6}=\frac{-4}{6}=-\frac{2}{3}\)

Answer:

\(y = 5\) or \(y=-\frac{2}{3}\)