Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.

Enter the correct answer in the box.

What are the solutions of this quadratic equation?
[tex]\[ x^2 = 16x - 65 \][/tex]

Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] to complete the solutions:
[tex]\[
\begin{array}{l}
x = a + b i \\
x = a - b i
\end{array}
\][/tex]


Sagot :

To find the solutions for the quadratic equation [tex]\( x^2 = 16x - 65 \)[/tex], we first need to rewrite it in standard form. The standard form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex].

Starting with the original equation:

[tex]\[ x^2 = 16x - 65 \][/tex]

We rearrange the terms to get:

[tex]\[ x^2 - 16x + 65 = 0 \][/tex]

Here, the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -16 \)[/tex]
- [tex]\( c = 65 \)[/tex]

To find the solutions, we use the quadratic formula:

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

First, we calculate the discriminant:

[tex]\[ \text{Discriminant} = b^2 - 4ac \][/tex]
[tex]\[ (-16)^2 - 4(1)(65) = 256 - 260 = -4 \][/tex]

Since the discriminant is negative, the equation has complex roots. Let's find the real and imaginary parts of these roots.

The real part of the roots is:

[tex]\[ \text{Real part} = \frac{-b}{2a} = \frac{16}{2} = 8 \][/tex]

The imaginary part of the roots is:

[tex]\[ \text{Imaginary part} = \frac{\sqrt{|\text{Discriminant}|}}{2a} = \frac{\sqrt{4}}{2} = 1 \][/tex]

So the solutions to the equation are in the form:

[tex]\[ x = a + bi \][/tex]
[tex]\[ x = a - bi \][/tex]

Substituting the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:

[tex]\[ x = 8 + 1i \][/tex]
[tex]\[ x = 8 - 1i \][/tex]

Therefore, the roots of the quadratic equation [tex]\( x^2 = 16x - 65 \)[/tex] are:

[tex]\[ x = 8 + 1i \][/tex]
[tex]\[ x = 8 - 1i \][/tex]