Use the quadratic formula to solve. Express your answer in simplest form.

Question

6, x, squared, minus, 5, x, minus, 2, equals, minus, 6, x, squared
6x
2
−5x−2=
−6x
2

Answer 1

To solve the equation $6x^2-5x-2=-6x^2$, we first need to bring all terms to one side of the equation.

1. Add $6x^2$ to both sides:

$$6x^2+6x^2-5x-2=0$$

This simplifies to:

$$12x^2-5x-2=0$$

2. Now we can apply the quadratic formula, which is given by:

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

Here, $a=12$, $b=-5$, and $c=-2$.

3. We first calculate the discriminant ($b^2-4ac$):

$$b^2-4ac=(-5{)}^2-4(12)(-2)=25+96=121$$

4. Now we substitute $a$, $b$, and the discriminant into the quadratic formula:

$$x=\frac{-(-5)\pm \sqrt{121}}{2(12)}$$

This simplifies to:

$$x=\frac{5\pm 11}{24}$$

5. Now we calculate the two possible values for $x$:

For the positive case:

$$x=\frac{5+11}{24}=\frac{16}{24}=\frac{2}{3}$$

For the negative case:

$$x=\frac{5-11}{24}=\frac{-6}{24}=-\frac{1}{4}$$

6. Therefore, the solutions to the equation $12x^2-5x-2=0$ are:

$$x=\frac{2}{3}\text{and}x=-\frac{1}{4}$$

These are the solutions in simplest form.