What are the roots of the equation 16x^2 = 56x -51 in simplest a + bi form

To find the roots of the equation \(16x^2 = 56x - 51\), we first rewrite it in standard quadratic form:

\[
16x^2 - 56x + 51 = 0
\]

Here, \(a = 16\), \(b = -56\), and \(c = 51\).

We will use the quadratic formula to find the roots:

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

First, let's calculate the discriminant \(D = b^2 - 4ac\):

\[
D = (-56)^2 - 4(16)(51)
\]

\[
D = 3136 - 3264
\]

\[
D = -128
\]

Since the discriminant \(D\) is negative, the roots will be complex.

Now, using the quadratic formula:

\[
x = \frac{-(-56) \pm \sqrt{-128}}{2 \cdot 16}
\]

\[
x = \frac{56 \pm \sqrt{-128}}{32}
\]

To simplify \(\sqrt{-128}\), we recognize that \(\sqrt{-128} = \sqrt{-1 \times 128} = i\sqrt{128}\).

Since \(\sqrt{128} = \sqrt{64 \times 2} = 8\sqrt{2}\), we have:

\[
\sqrt{-128} = 8i\sqrt{2}
\]

Substituting this back into the formula gives:

\[
x = \frac{56 \pm 8i\sqrt{2}}{32}
\]

We can simplify this by dividing both terms by 8:

\[
x = \frac{56}{32} \pm \frac{8i\sqrt{2}}{32}
\]

\[
x = \frac{7}{4} \pm \frac{i\sqrt{2}}{4}
\]

Therefore, the roots of the equation \(16x^2 = 56x - 51\) in the form \(a + bi\) are:

\[
x_1 = \frac{7}{4} + \frac{i\sqrt{2}}{4}, \quad x_2 = \frac{7}{4} - \frac{i\sqrt{2}}{4}
\]

These can also be written as:

\[
x_1 = 1.75 + 0.25i\sqrt{2}, \quad x_2 = 1.75 - 0.25i\sqrt{2}
\]

Would you like more details on any step, or do you have any questions?

Here are some related questions you might be interested in:
1. What is the significance of the discriminant in determining the nature of the roots?
2. How can we express complex numbers in polar form?
3. What are conjugate pairs, and why do complex roots always come in pairs?
4. How can complex roots be represented graphically on the complex plane?
5. Can the quadratic formula be used to solve equations with complex coefficients?

**Tip:** When simplifying square roots of negative numbers, remember to separate the negative part as \(i\) and focus on simplifying the positive square root first.

Learn how to find the roots of the quadratic equation 16x^2 = 56x - 51 in simplest a + bi form. This problem covers quadratic equations, complex numbers, and the quadratic formula. Suitable for high school students.

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