Solve the quadratic equation 4x^2 + 2x - 3 = 0
Answers:
To solve the quadratic equation ( 4x^2 + 2x - 3 = 0 ), we can use the quadratic formula:
[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]
where ( a = 4 ), ( b = 2 ), and ( c = -3 ).
First, we need to calculate the discriminant ( b^2 - 4ac ):
[ b^2 - 4ac = (2)^2 - 4(4)(-3) = 4 + 48 = 52 ]
Next, we substitute ( a ), ( b ), and the discriminant into the quadratic formula:
[ x = \frac{-2 \pm \sqrt{52}}{2 \cdot 4} ]
We can simplify ( \sqrt{52} ):
[ \sqrt{52} = \sqrt{4 \cdot 13} = 2\sqrt{13} ]
Now substituting back into the formula:
[ x = \frac{-2 \pm 2\sqrt{13}}{8} ]
This simplifies to:
[ x = \frac{-1 \pm \sqrt{13}}{4} ]
Thus, the solutions to the quadratic equation ( 4x^2 + 2x - 3 = 0 ) are:
[ x = \frac{-1 + \sqrt{13}}{4} \quad \text{and} \quad x = \frac{-1 - \sqrt{13}}{4} ]