Expert: Scott A Wilson - 2/8/2010

Question
Hello, i need help with algebra homework.

Solve for x in terms of y.

3x^2+ax^2=9x+9a

6a^2*x^2-11ax=10


Solve by completing the square

x^2=(6 square root 2)x+7


Please help

Answer
Is see no y in the equations 3x^2+ax^2=9x+9a and 6a^2*x^2-11ax=10.

If you mean solve for a, then you do that in eac equation.
On the 1st equation, subtract 3x² from both sides and subtract 9a from both sides.
This gives ax² - 9a = 9x - 3x².

Factor out the a on the left and 3x on the right.  Thig gives a(x²-9) = 3x(3-x).

Note that x² - 9 = (x+3)(x-3) and note that 3-x is -(x-3).

Divide both sides of the equation by x-3 and get a(x+3) = -3x.

Divide both sides by x+3 and get a = -3x/(x+3).

I would then put the into the 2nd equation.  Note on the second term, the - signs cancel.
This gives 6(9x²/(x+3)²)x² + 11(3x/(x+3))x = 10.

Multilpy bt (x+3)² and get 54x^4 + 33x²(x+3) = 10(x+3)².

Multiplying each term gives 54x^4 + 33x³ + 99x² = 10x² + 60x + 90.

Subtracting the right side form both sides gives  54x^4 + 33x³ + 89x² - 60x - 90 = 0.

I would then put this equation in Excel and find the roots.
Put -10 in A1 and put the formula in B1.
Put =A1+1 in A2 and copy B1 to B2.
Highlight A2:B20 and do an Edit-Fill-Down.

Doing so once gave me
-1   80
-0.9   47.4624
-0.8   20.1824
-0.7   -2.7436
-0.6   -22.0896
-0.5   -38.5
-0.4   -52.4896
-0.3   -64.4436
-0.2   -74.6176
-0.1   -83.1376
0.0   -90
0.1   -95.0716
0.2   -98.0896
0.3   -98.6616
0.4   -96.2656
0.5   -90.25
0.6   -79.8336
0.7   -64.1056
0.8   -42.0256
0.9   -12.4236
1   26

By trial and error, I would find the two roots.
One of them would be between -0.8 and -0.7 and the other would be between 0.9 and 1.

For x² = (6√2)x + 7, subtract (6√2)x from both sides.  This gives x² - 6√2x = 7.

Take half of 6√2, giving 3√2, and then square, giving 9*2 = 18.

Add this to both side, so you have x² - 6√2x + 18 = 25.

The equation can now be factored into (x-3√2)² = 5².