Questions: Which of the following equations are equivalent to ax^2+bx+c=0? x=-b ± sqrt(b^2-4ac)/2a x^2+bx=-c (x+b/(2a))^2=(b^2-4ac)/(2a) (x+b/(2a))^2=(b^2-4ac)/(4a^2) x^2+(b/a)x+(c/a)=0 x=(-b ± sqrt(b^2-4ac))/(2a) ax^2+bx=-c

Which of the following equations are equivalent to ax^2+bx+c=0?
x=-b ± sqrt(b^2-4ac)/2a
x^2+bx=-c
(x+b/(2a))^2=(b^2-4ac)/(2a)
(x+b/(2a))^2=(b^2-4ac)/(4a^2)
x^2+(b/a)x+(c/a)=0
x=(-b ± sqrt(b^2-4ac))/(2a)
ax^2+bx=-c

Transcript text: Which of the following equations are equivalent to $a x^{2}+b x+c=0$ ? $x=-b \pm \frac{\sqrt{b^{2}-4 a c}}{2 a}$ $x^{2}+b x=-c$ $\left(x+\frac{b}{2 a}\right)^{2}=\frac{b^{2}-4 a c}{2 a}$ $\left(x+\frac{b}{2 a}\right)^{2}=\frac{b^{2}-4 a c}{4 a^{2}}$ $x^{2}+\frac{b}{a} x+\frac{c}{a}=0$ $x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}$ $a x^{2}+b x=-c$

Solution

Solution Steps

To determine which of the given equations are equivalent to the quadratic equation $ ax^2 + bx + c = 0 $, we need to analyze each option and see if it can be derived from or transformed into the standard quadratic form.

The quadratic formula $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $ is a direct solution to the quadratic equation.
Rearranging $ x^2 + bx = -c $ to $ x^2 + bx + c = 0 $ shows it is equivalent.
Completing the square on $ ax^2 + bx + c = 0 $ can lead to the form $ \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} $.
The equation $ x^2 + \frac{b}{a}x + \frac{c}{a} = 0 $ is obtained by dividing the original equation by $ a $.

Step 1: Analyze the Given Equations

We need to determine which of the given equations are equivalent to the quadratic equation $ ax^2 + bx + c = 0 $.

Step 2: Quadratic Formula

The quadratic formula $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $ is a direct solution to the quadratic equation. This matches the given equation: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Step 3: Rearrange $ x^2 + bx = -c $

Rearranging $ x^2 + bx = -c $ to $ x^2 + bx + c = 0 $ shows it is equivalent to the original quadratic equation.

Step 4: Completing the Square

Completing the square on $ ax^2 + bx + c = 0 $ can lead to the form: \[ \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} \]

Step 5: Dividing by $ a $

The equation $ x^2 + \frac{b}{a}x + \frac{c}{a} = 0 $ is obtained by dividing the original equation by $ a $.

Step 6: Verify Equivalence

Based on the Python output, the following equations are equivalent to $ ax^2 + bx + c = 0 $:

$ \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} $
$ x^2 + \frac{b}{a}x + \frac{c}{a} = 0 $
$ ax^2 + bx = -c $

Final Answer

The equivalent equations to $ ax^2 + bx + c = 0 $ are: \[ \boxed{\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}} \] \[ \boxed{x^2 + \frac{b}{a}x + \frac{c}{a} = 0} \] \[ \boxed{ax^2 + bx = -c} \]

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