x²-x-1=0 Solve by completing the square?

Question

Solve by completing the square.
1) x²-x-1=0

Use the quadratic formula to solve the following. Leave irrational roots in the simplest radical form.
3x²-4x-5=0

Answer 1

x² - x - 1 = 0
x² - x + 1/4 = 1 + 1/4
(x - 1/2)² = 5/4
x - 1/2 = ±(√5)/2
x = (1 ± √5)/2

3x² - 4x - 5 = 0
x = 4/6 ± (√(16 + 60))/6
x = 2/3 ± (√19)/3

Answer 2

winding up the sq. potential which you relatively decide to have the capacity to place in writing your quadratic, ax^2 + bx + c = 0, contained in the form (Ax + B)^2 = C If we expand that form, we get A^2 x^2 + 2AB x + B^2 = C So, you opt to function some consistent to the two factors of the equation so as which you get a sort which you will decrease. If we circulate C decrease back over to the left, we get A^2 x^2 + 2AB x + B^2 - C = 0 Now, comparing this to our unique quadratic, we've A^2 = a 2AB = b B^2 - C = c A = ?a B = b / (2A) B = b / (2?a) C = B^2 - c C = (b / (2?a))^2 - c C = b^2 / (4a) - c on your equation, x^2 + 10x - a million = 0 you have a = a million, b = 10, and c = -a million. So C = 10^2 / (4*a million) - -a million C = a hundred / 4 + a million C = 25 + a million C = 26 So, we finished the sq. by potential of including 26 to the two factors, which aspects us x^2 + 10x - a million + 26 = 26 x^2 + 10x + 25 = 26 Now, we can element that into (x + 5)^2 = 26 (of direction, if we remembered that A = ?a, and B = b / (2?a), we could pass the factoring.) And from there, this is worry-free to sparkling up. x + 5 = ±?26 x = -5 ±?26 Now, it is not likely which you are going to undergo in ideas that, or decide to derive it. So the easy thank you to undergo in ideas what to do is, divide by potential of the coefficient of the x^2 term. Then, you like the consistent term to be equivalent to a million/2 the coefficient of x, squared. So upload regardless of you need to the two factors to make that take place. to illustrate, in case you have 3x^2 + 18x + 12 = 0 ... divide by potential of three x^2 + 6x + 4 = 0 ... a million/2 of 6 is 3; 3^2 is 9. So, what do we upload to 4 to make 9? that could be 5. x^2 + 6x + 4 + 5 = 5 x^2 + 6x + 9 = 5 ... now element into the sq. of a linear expression (undergo in ideas, the consistent would be a million/2 of the coefficient on the x term contained in the quadratic) (x + 3)^2 = 5 Or, in case you had 5x^2 - 45x - 35 = 0 ... divide by potential of 5 x^2 - 9x - 7 = 0 ... a million/2 of -9 is -9/2; (-9/2)^2 = 80 one/4. What do we upload to -7 to make 80 one/4? -7 = -28/4; 80 one/4 - -28/4 = (80 one + 28) / 4 = 109/4 x^2 - 9x + 80 one/4 = 109/4 ... element (x - 9/2)^2 = 109/4 i wish that facilitates!

Answer 3

x^2-x-1=0
x^2-x=1
x^2-x+1/4=1+1/4
(x-1/2)^2=5/4
x-1/2=+-Sqrt(5/4)
x=1/2+-Sqrt(5/4).

3x^2-4x-5=0

x=(4+-Sqrt(16-4(3)(-5)))/(2)(3)
x=(4+-Sqrt(16+60))/6
x=(4+-Sqrt(76))/6