The equation y=x2+12ax+a represents a parabola for all real values of a

Let a, b∈R and a≠b. Show that the equation x2+12ax+a=x2+12bx+b has the unique solution x=−2.

Explained: "The equation y=x2+12ax+a represents a parabola for all..."

High School
Algebra
Algebra I
Quadratic function and equation

Sereinserenormg

Answered question

2022-01-30

The equation

y = x^{2} + \frac{1}{2} a x + a

represents a parabola for all real values of a

Answer & Explanation

trovabile4p

Beginner2022-01-31Added 13 answers

Let a,

b \in R and a \neq b

. Show that the equation

x^{2} + \frac{1}{2} a x + a = x^{2} + \frac{1}{2} b x + b

has the unique solution

x = - 2

Gordon Stephens

Beginner2022-02-01Added 10 answers

Just a disclaimer this is not the most formal proof so take it with some skepticism
Considered rearranging the quadratic so that a is the subject. You get

a = \frac{y - x^{2}}{\frac{x}{2} + 1}

since a can take any real value then the only candidates for x and y must make the right hand side an indeterminate form, e.g. since x and y are both finite both neumerator and denominator must be zero. So we want

\frac{x}{2} + 1 = 0

which gives

x = - 2 and y - x^{2} = 0

which gives

y = 4

. All that's left is to substitute these values into the original equation to verify that this is indeed a shared point.

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