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Select the correct answer.

Consider this quadratic equation:
[tex]\[ 2x^2 - 1 = 3x + 4 \][/tex]

Which equation correctly applies the quadratic formula?

A. [tex]\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(-5)}}{2} \][/tex]

B. [tex]\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(-5)}}{(2)} \][/tex]

C. [tex]\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - (2)(-5)}}{(2)} \][/tex]

D. [tex]\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(-5)}}{2(2)} \][/tex]


Sagot :

To determine the correct equation that applies the quadratic formula, we need to start by rearranging the given quadratic equation [tex]\(2x^2 - 1 = 3x + 4\)[/tex] into the standard form [tex]\(ax^2 + bx + c = 0\)[/tex].

Here are the steps:

1. Start with the original equation:
[tex]\[ 2x^2 - 1 = 3x + 4 \][/tex]

2. Move all terms to one side to bring it to the standard form:
[tex]\[ 2x^2 - 1 - 3x - 4 = 0 \][/tex]

3. Simplify the equation:
[tex]\[ 2x^2 - 3x - 5 = 0 \][/tex]

Now we have the quadratic equation in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex], where [tex]\(a = 2\)[/tex], [tex]\(b = -3\)[/tex], and [tex]\(c = -5\)[/tex].

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

Substitute [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the formula:

- [tex]\(a = 2\)[/tex]
- [tex]\(b = -3\)[/tex]
- [tex]\(c = -5\)[/tex]

This gives us:
[tex]\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot (-5)}}{2 \cdot 2} \][/tex]

Simplify the expression further:

- The numerator should be:
[tex]\[ -(-3) = 3 \][/tex]

- The discriminant inside the square root:
[tex]\[ (-3)^2 - 4 \cdot 2 \cdot (-5) = 9 + 40 = 49 \][/tex]

- The denominator:
[tex]\[ 2 \cdot 2 = 4 \][/tex]

Thus, the correct equation applying the quadratic formula is:
[tex]\[ x = \frac{3 \pm \sqrt{49}}{4} \][/tex]

Therefore, the answer choice that correctly applies the quadratic formula is:

D. [tex]\(x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot (-5)}}{2 \cdot 2}\)[/tex]