Select the correct answer.

Consider the quadratic equation below:
[tex]\[ 4x^2 - 5 = 3x + 4 \][/tex]

Determine the correct set-up for solving the equation using the quadratic formula.

A. [tex]\[ x = \frac{-(3) \pm \sqrt{(3)^2 - 4(-4)(1)}}{2(1)} \][/tex]

B. [tex]\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(4)(9)}}{2(4)} \][/tex]

C. [tex]\[ x = \frac{-(3) \pm \sqrt{(3)^2 - 4(-4)(-9)}}{2(-4)} \][/tex]

D. [tex]\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(4)(-9)}}{2(4)} \][/tex]



Answer :

First, let's rearrange the given quadratic equation into the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex]. We start with:

[tex]\[ 4x^2 - 5 = 3x + 4 \][/tex]

To convert this into standard form, move all terms to one side of the equation:

[tex]\[ 4x^2 - 5 - 3x - 4 = 0 \][/tex]

Combine like terms:

[tex]\[ 4x^2 - 3x - 9 = 0 \][/tex]

Now it's in standard form, where [tex]\(a = 4\)[/tex], [tex]\(b = -3\)[/tex], and [tex]\(c = -9\)[/tex].

The quadratic formula is given by:

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

1. Identify the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] again:
- [tex]\(a = 4\)[/tex]
- [tex]\(b = -3\)[/tex]
- [tex]\(c = -9\)[/tex]

2. Substitute these values into the quadratic formula:
- The term [tex]\(-b\)[/tex] becomes [tex]\(-(-3) = 3\)[/tex].
- The term [tex]\(b^2\)[/tex] is [tex]\((-3)^2 = 9\)[/tex].
- The term [tex]\(4ac\)[/tex] is [tex]\(4 \cdot 4 \cdot -9 = -144\)[/tex].

Putting it all together, the discriminant [tex]\( \Delta \)[/tex] is:

[tex]\[ \Delta = b^2 - 4ac = 9 - (-144) = 9 + 144 = 153 \][/tex]

3. So, the quadratic formula setup becomes:

[tex]\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 4 \cdot (-9)}}{2 \cdot 4} \][/tex]

The correct setup is:

[tex]\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 4 \cdot (-9)}}{2 \cdot 4} \][/tex]

Therefore, the correct answer is:

D. [tex]\( x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(4)(-9)}}{2(4)} \)[/tex]

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