Question
Use the quadratic formula to solve. Express your answer in simplest form.
6, x, squared, minus, 5, x, minus, 2, equals, minus, 6, x, squared
6x
2
−5x−2=
−6x
2
6, x, squared, minus, 5, x, minus, 2, equals, minus, 6, x, squared
6x
2
−5x−2=
−6x
2
Answers
There are no human answers yet.
Answered by
GPT-4o mini
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:
\[
x = \frac{2}{3} \quad \text{and} \quad x = -\frac{1}{4}
\]
These are the solutions in simplest form.
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:
\[
x = \frac{2}{3} \quad \text{and} \quad x = -\frac{1}{4}
\]
These are the solutions in simplest form.
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