Algebra

3x^2+x-1/3=0

Step-by-step solution with explanation

Final Answer

and

Step-by-step solution

1

Write the equation in standard form

We have a quadratic equation already in standard form . We identify , , and .
2

Clear the fraction by multiplying through

Multiply every term by 3 to eliminate the fraction. This gives us , , , which are easier to work with.
3

Write the quadratic formula

Since this quadratic doesn't factor neatly, we use the quadratic formula. Plug in , , .
4

Calculate the discriminant

The discriminant tells us how many solutions exist. Since , there are two real solutions.
5

Simplify the square root

Factor 45 into a perfect square times 5 so we can simplify the radical.
6

Substitute into the quadratic formula

Plug the discriminant result back into the formula. Now simplify the fraction.
7

Simplify and state both solutions

Divide numerator and denominator by 3. This gives two solutions: and .

Understanding this problem

Learning Insight

Multiplying through by a number to clear fractions doesn't change the solutions — it just makes the numbers easier to handle. The quadratic formula always works for any as long as , because it is derived directly from completing the square.

Quick Tip

When you see a quadratic with a fraction, multiply every term by the denominator first. It turns a messy problem into a clean one without changing the answers.

Common Mistake

Students often forget to multiply ALL terms (including the constant) when clearing fractions, turning into but leaving other terms unchanged — always multiply every single term.