Question
Solve the equation for all real solutions in simplest form.
4, m, squared, minus, 5, m, minus, 3, equals, 2, m
4m
2
−5m−3=
2m
4, m, squared, minus, 5, m, minus, 3, equals, 2, m
4m
2
−5m−3=
2m
Answers
GPT-4o mini
To solve the equation \( 4m^2 - 5m - 3 = 2m \), we first rearrange the equation to set it to zero.
We can do this by subtracting \( 2m \) from both sides:
\[
4m^2 - 5m - 3 - 2m = 0
\]
This simplifies to:
\[
4m^2 - 7m - 3 = 0
\]
Next, we can solve this quadratic equation using the quadratic formula:
\[
m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
In our case, \( a = 4 \), \( b = -7 \), and \( c = -3 \). We first calculate the discriminant:
\[
b^2 - 4ac = (-7)^2 - 4(4)(-3) = 49 + 48 = 97
\]
Now we can plug the values into the quadratic formula:
\[
m = \frac{-(-7) \pm \sqrt{97}}{2 \cdot 4}
\]
This simplifies to:
\[
m = \frac{7 \pm \sqrt{97}}{8}
\]
Thus, the two solutions for \( m \) are:
\[
m = \frac{7 + \sqrt{97}}{8} \quad \text{and} \quad m = \frac{7 - \sqrt{97}}{8}
\]
These are the real solutions in simplest form.
We can do this by subtracting \( 2m \) from both sides:
\[
4m^2 - 5m - 3 - 2m = 0
\]
This simplifies to:
\[
4m^2 - 7m - 3 = 0
\]
Next, we can solve this quadratic equation using the quadratic formula:
\[
m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
In our case, \( a = 4 \), \( b = -7 \), and \( c = -3 \). We first calculate the discriminant:
\[
b^2 - 4ac = (-7)^2 - 4(4)(-3) = 49 + 48 = 97
\]
Now we can plug the values into the quadratic formula:
\[
m = \frac{-(-7) \pm \sqrt{97}}{2 \cdot 4}
\]
This simplifies to:
\[
m = \frac{7 \pm \sqrt{97}}{8}
\]
Thus, the two solutions for \( m \) are:
\[
m = \frac{7 + \sqrt{97}}{8} \quad \text{and} \quad m = \frac{7 - \sqrt{97}}{8}
\]
These are the real solutions in simplest form.