Use the quadratic formula to solve the quadratic equation

8
x
2
+
5
=
12
x
. Express its solutions in the form
a
±
b
i
.(1 point)

1 answer

To solve the quadratic equation \( 8x^2 + 5 = 12x \), we first rearrange it into standard form. We do this by moving all terms to one side of the equation:

\[ 8x^2 - 12x + 5 = 0 \]

Now that we have it in the standard form \( ax^2 + bx + c = 0 \), we can identify the coefficients:

  • \( a = 8 \)
  • \( b = -12 \)
  • \( c = 5 \)

Next, we use the quadratic formula:

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

Now we calculate the discriminant \( b^2 - 4ac \):

\[ b^2 = (-12)^2 = 144 \] \[ 4ac = 4 \times 8 \times 5 = 160 \] \[ b^2 - 4ac = 144 - 160 = -16 \]

Since the discriminant is negative, the solutions will be complex numbers. We can now substitute the values into the quadratic formula:

\[ x = \frac{-(-12) \pm \sqrt{-16}}{2 \times 8} \] \[ x = \frac{12 \pm \sqrt{-16}}{16} \] \[ x = \frac{12 \pm 4i}{16} \]

We can simplify this expression:

\[ x = \frac{12}{16} \pm \frac{4i}{16} \] \[ x = \frac{3}{4} \pm \frac{i}{4} \]

Thus, the solutions of the quadratic equation \( 8x^2 + 5 = 12x \) in the form \( a \pm bi \) are:

\[ x = \frac{3}{4} \pm \frac{1}{4}i \]