Let's tackle the derivation of the quadratic formula step-by-step and fill in the missing steps with the appropriate given choices:
Step 1: [tex]\( a x^2 + b x + c = 0 \)[/tex]
This is the general form of the quadratic equation.
Step 2: [tex]\( a x^2 + b x = -c \)[/tex]
We move the constant term [tex]\( c \)[/tex] to the other side of the equation.
Step 3: [tex]\( B: x^2 + \frac{b}{a} x = \frac{-c}{a} \)[/tex]
We divide every term by [tex]\( a \)[/tex] to normalize the coefficient of [tex]\( x^2 \)[/tex] to 1.
Step 4: [tex]\( x^2 + \frac{b}{a} x + \frac{b^2}{4 a^2} = \frac{-c}{a} + \frac{b^2}{4 a^2} \)[/tex]
We add [tex]\(\frac{b^2}{4 a^2}\)[/tex] to both sides of the equation to complete the square on the left side.
Step 5: [tex]\( D: \left(x + \frac{b}{2 a} \right)^2 = \frac{b^2}{4 a^2} - \frac{4 a c}{4 a^2} \)[/tex]
We complete the square on the left and combine the fractions on the right.
Step 6: [tex]\( (x + \frac{b}{2 a})^2 = \frac{b^2 - 4 a c}{4 a^2} \)[/tex]
We simplify the right-hand side.
Step 7: [tex]\( x = -\frac{b}{2 a} \pm \frac{\sqrt{b^2 - 4 a c}}{2 a} \)[/tex]
Taking the square root of both sides, we solve for [tex]\(x\)[/tex].
Step 8: [tex]\( C: x = \frac{-b \pm \sqrt{b^2 - 4 a c}}{2 a} \)[/tex]
This is the simplified form of the quadratic formula.
Therefore, the filled steps are:
1. [tex]\( a x^2 + b x + c = 0 \)[/tex]
2. [tex]\( a x^2 + b x = -c \)[/tex]
3. [tex]\( B: x^2 + \frac{b}{a} x = \frac{-c}{a} \)[/tex]
4. [tex]\( x^2 + \frac{b}{a} x + \frac{b^2}{4 a^2} = \frac{-c}{a} + \frac{b^2}{4 a^2} \)[/tex]
5. [tex]\( D: \left(x + \frac{b}{2 a} \right)^2 = \frac{b^2}{4 a^2} - \frac{4 a c}{4 a^2} \)[/tex]
6. [tex]\( (x + \frac{b}{2 a})^2 = \frac{b^2 - 4 a c}{4 a^2} \)[/tex]
7. [tex]\( x = -\frac{b}{2 a} \pm \frac{\sqrt{b^2 - 4 a c}}{2 a} \)[/tex]
8. [tex]\( C: x = \frac{-b \pm \sqrt{b^2 - 4 a c}}{2 a} \)[/tex]
This completes the derivation of the quadratic formula with the correctly filled steps.
