Algebra Know-It-ALL

and

3 x^2 +y+ 5 x− 11 = 0

First, we morph

In both of these equations, a multiple of y can be separated out and placed alone on the left

side of the equals sign, producing quadratic functions of x. In the first equation, we can sub-

tract 2y from each side and then transpose the sides to get

− 2 y= 4 x^2 + 6 x+ 8

Dividing through by −2, we get the function

y=− 2 x^2 − 3 x− 4

In the second original equation, we can subtract y from each side and then transpose the sides,

getting

−y= 3 x^2 + 5 x− 11

Multiplying through by −1, we get

y=− 3 x^2 − 5 x+ 11

Next, we mix

When we directly mix the right sides of the above two quadratic functions, we get a single

equation in one variable:

− 2 x^2 − 3 x− 4 =− 3 x^2 − 5 x+ 11

We can add the quantity (3x^2 + 5 x− 11) to each side, obtaining

x^2 + 2 x− 15 = 0

which is a quadratic equation in polynomial standard form.

Next, we solve

We now have an equation that can be easily factored. It does not take long to figure out that

the above quadratic is equivalent to

(x+ 5)(x− 3) = 0

The roots are found by solving the equations

x+ 5 = 0

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