Answer:
The correct next step is the answer choice
[tex]x^{2} +\frac{b}{a}x=-\frac{c}{a}[/tex]
Step-by-step explanation:
we have the quadratic equation in standard form
[tex]ax^{2}+bx+c=0[/tex]
The steps in the derivation of the quadratic formula by completing the square are
step 1
[tex]ax^{2}+bx+c=0[/tex] ----> given equation
step 2
Move the constant term over to the right-hand side
[tex]ax^{2}+bx=-c[/tex]
step 3
The leading term is multiplied by a
so
Divide by a both sides
[tex]x^{2} +\frac{b}{a}x=-\frac{c}{a}[/tex]
step 4
Multiply the linear term by 1/2
[tex]\frac{b}{a}(\frac{1}{2})=+\frac{b}{2a}[/tex]
step 5
square this derived value
[tex]+\frac{b^2}{4a^2}[/tex]
step 6
Add this squared value to either side of the equation
[tex]x^{2} +\frac{b}{a}x+\frac{b^2}{4a^2}=-\frac{c}{a}+\frac{b^2}{4a^2}[/tex]
step 7
convert to the common denominator, and combine on the right-hand side
[tex]x^{2} +\frac{b}{a}x+\frac{b^2}{4a^2}=\frac{b^2-4ac}{4a^2}[/tex]
step 8
convert the left-hand side to completed-square form
[tex](x+\frac{b}{2a})^{2}=\frac{b^2-4ac}{4a^2}[/tex]
step 9
take the square roots of either side
[tex](x+\frac{b}{2a})=(+/-)\sqrt{\frac{b^2-4ac}{4a^2}} \\\\(x+\frac{b}{2a})=(+/-)\frac{\sqrt{b^2-4ac}}{2a}[/tex]
step 10
solving for the variable
[tex]x=-\frac{b}{2a}(+/-)\frac{\sqrt{b^2-4ac}}{2a}\\\\x=\frac{-b(+/-)\sqrt{b^2-4ac}}{2a}[/tex]
