To solve the quadratic equation [tex]\( 0 = x^2 - 2x - 3 \)[/tex] using the quadratic formula, we need to substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] appropriately.
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \][/tex]
For the equation [tex]\( x^2 - 2x - 3 = 0 \)[/tex], we identify the coefficients as follows:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -2 \)[/tex]
- [tex]\( c = -3 \)[/tex]
Let's look at the given options and check which one shows the correct substitution.
1. [tex]\[ \frac{2 \pm \sqrt{(-2)^2 - 4(1)(-3)}}{2(1)} \][/tex]
2. [tex]\[ \frac{-2 \pm \sqrt{(-2)^2 - 4(1)(-3)}}{2(1)} \][/tex]
3. [tex]\[ \frac{2 \pm \sqrt{-2^2 - 4(1)(-3)}}{2(1)} \][/tex]
4. [tex]\[ \frac{-2 \pm \sqrt{-2^2 - 4(1)(-3)}}{2(1)} \][/tex]
Let's analyze each option:
1. In this option, the numerator starts with [tex]\( 2 \)[/tex] instead of [tex]\(-b\)[/tex]. Since [tex]\( b = -2 \)[/tex], [tex]\(-b\)[/tex] should be [tex]\( -(-2) = 2 \)[/tex]. Therefore, although the square root part is correct, starting with [tex]\( 2 \)[/tex] is incorrect.
2. This option starts with [tex]\( -2 \)[/tex], which corresponds correctly to [tex]\( b \)[/tex]. Hence, it follows [tex]\( -b = -(-2) \)[/tex]. The entire formula here is correctly substituted.
3. This option again starts with [tex]\( 2 \)[/tex] but uses [tex]\( -2^2 \)[/tex], which should be in parentheses [tex]\( (-2)^2 \)[/tex]. Therefore, this is incorrect.
4. This option starts with [tex]\( -2 \)[/tex], which is correct, but the square root term [tex]\( -2^2 \)[/tex] should be [tex]\( (-2)^2 \)[/tex]. Hence, this is also incorrect.
The correct substitution according to the quadratic formula should include [tex]\(-b\)[/tex]. Therefore, the correct substitution is:
[tex]\[ \frac{-2 \pm \sqrt{ (-2)^2 - 4(1)(-3) }}{ 2(1) }. \][/tex]
So, the correct answer is:
[tex]\[ \frac{-2 \pm \sqrt{(-2)^2 - 4(1)(-3)}}{2(1)} \][/tex]
