Welcome to Westonci.ca, your one-stop destination for finding answers to all your questions. Join our expert community now! Join our Q&A platform to connect with experts dedicated to providing precise answers to your questions in different areas. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.

Consider this quadratic equation:

[tex]\(x^2 + 1 = 2x - 3\)[/tex]

Which expression correctly sets up the quadratic formula?

A. [tex]\(\frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(4)}}{2(1)}\)[/tex]

B. [tex]\(\frac{-(-2) \pm \sqrt{(-2)^2 - (1)(4)}}{2(2)}\)[/tex]

C. [tex]\(\frac{-2 \pm \sqrt{(-2)^2 - 4(1)(4)}}{2(1)}\)[/tex]

D. [tex]\(\frac{-2 \pm \sqrt{(2)^2 - 4(1)(-2)}}{2(1)}\)[/tex]


Sagot :

To solve the quadratic equation [tex]\(x^2 + 1 = 2x - 3\)[/tex], we need to first bring all terms to one side of the equation to have it in standard form:
[tex]\[x^2 + 1 - 2x + 3 = 0.\][/tex]

Simplifying the equation:
[tex]\[x^2 - 2x + 4 = 0.\][/tex]

This is a standard quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] where [tex]\(a = 1\)[/tex], [tex]\(b = -2\)[/tex], and [tex]\(c = 4\)[/tex].

The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \][/tex]

We need to find the correct expression that sets up the quadratic formula. Let's analyze each option:

A. [tex]\(\frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(4)}}{2(1)}\)[/tex]

B. [tex]\(\frac{-(-2) \pm \sqrt{(-2)^2 - (1)(4)}}{2(2)}\)[/tex]

C. [tex]\(\frac{-2 \pm \sqrt{(-2)^2 - 4(1)(4)}}{2(1)}\)[/tex]

D. [tex]\(\frac{-2 \pm \sqrt{(2)^2 - 4(1)(-2)}}{2(1)}\)[/tex]

Step-by-step Analysis:

1. Substitute [tex]\(a = 1\)[/tex], [tex]\(b = -2\)[/tex], and [tex]\(c = 4\)[/tex] into the quadratic formula:
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(4)}}{2(1)}. \][/tex]

2. Simplify within the discriminant (under the square root):
[tex]\[ (-2)^2 = 4, \][/tex]
[tex]\[ 4ac = 4 \times 1 \times 4 = 16, \][/tex]
[tex]\[ b^2 - 4ac = 4 - 16 = -12. \][/tex]

3. Substitute back into the formula:
[tex]\[ x = \frac{2 \pm \sqrt{-12}}{2}. \][/tex]

Confirming Results:

- Option A:
[tex]\[ \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(4)}}{2(1)} \][/tex]
[tex]\[ \frac{2 \pm \sqrt{4 - 16}}{2} = \frac{2 \pm \sqrt{-12}}{2} \][/tex]
This matches our calculation.

- Option B:
[tex]\[ \frac{-(-2) \pm \sqrt{(-2)^2 - (1)(4)}}{2(2)} \][/tex]
This has an incorrect explanation within the discriminant [tex]\((-2)^2 - (1)(4) = 4 - 4 = 0\)[/tex] and wrong denominator [tex]\(2(2)\)[/tex].

- Option C:
[tex]\[ \frac{-2 \pm \sqrt{(-2)^2 - 4(1)(4)}}{2(1)} \][/tex]
This does not match correctly as it uses [tex]\( -2 \)[/tex] instead of [tex]\( 2 \)[/tex].

- Option D:
[tex]\[ \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-2)}}{2(1)} \][/tex]
This contains [tex]\( -4 \)[/tex].\ ([tex]\(2)^2 - 4(1)(-2) = 4 + 8 = 12\)[/tex]

Thus, the correct expression is:

[tex]\(\boxed{\frac{2 \pm \sqrt{-12}}{2}}\)[/tex]

Therefore, the correct expression that sets up the quadratic formula is:
[tex]\[ \boxed{A. \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(4)}}{2(1)}} \][/tex]

Thus, the correct answer is [tex]\(1\)[/tex].
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.