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Rhett is solving the quadratic equation [tex]\(0 = x^2 - 2x - 3\)[/tex] using the quadratic formula. Which shows the correct substitution of the values [tex]\(a, b\)[/tex], and [tex]\(c\)[/tex] into the quadratic formula?

Quadratic formula: [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex]

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

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

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

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


Sagot :

To solve the quadratic equation [tex]\(0 = x^2 - 2x - 3\)[/tex] using the quadratic formula, we need to correctly substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the formula. The quadratic formula is given by:

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

Here, the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are taken from the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex].

For the equation [tex]\(0 = x^2 - 2x - 3\)[/tex]:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -2\)[/tex]
- [tex]\(c = -3\)[/tex]

We will now substitute these values into the quadratic formula step-by-step:

1. Put the value of [tex]\(b\)[/tex] into the formula:
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-3)}}{2 \cdot 1} \][/tex]

2. Simplify inside the square root and fraction:
[tex]\[ x = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-3)}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{4 + 12}}{2} \][/tex]

3. Combine the terms inside the square root:
[tex]\[ x = \frac{2 \pm \sqrt{16}}{2} \][/tex]

4. Taking the square root of 16:
[tex]\[ x = \frac{2 \pm 4}{2} \][/tex]

Given the original options, the ones that match this process are:
[tex]\[ \frac{-2 \pm \sqrt{(-2)^2 - 4(1)(-3)}}{2(1)} \][/tex]
and
[tex]\[ \frac{-2 \pm \sqrt{-2^2 - 4(1)(-3)}}{2(1)} \][/tex]

Among these choices, while they appear similar, only the following shows the fully simplified version and correct signs:
[tex]\[ \frac{-2 \pm \sqrt{(-2)^2 - 4(1)(-3)}}{2(1)} \][/tex]

Thus, the correct substitution of the values [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the quadratic formula is:
[tex]\[ \boxed{\frac{-2 \pm \sqrt{(-2)^2 - 4(1)(-3)}}{2(1)}} \][/tex]