If: x -y = 2 then x = 2 +y
If: x^2 -4y^2 = 5 then (2 +y)^2 -4y^2 -5 = 0
Removing brackets: 4 +4y +y^2 -4y^2 -5 = 0
Collecting like terms: 4y -3y^2 -1 = 0
Divide all terms by -3: y^2 -4/3y +1/3 = 0
Completing the square: (y -2/3)^2 -4/9 +1/3 = 0 => (y -2/3)^2 = 1/9
Square root both sides: y -2/3 = -1/3 or +1/3
Add 2/3 to both sides: y = 1/3 or 1
By substitution into original equation intersection are at: (7/3, 1/3) and (3, 1)
x - y = 2 => x = y + 2Then x2 - 4y2 = 5 => (y + 2)2 - 4y2 = 5 => y2 + 4y + 4 - 4y2 = 5 => 3y2 - 4y + 1 = 0
Dividing by 3 gives y2 - (4/3)y + 1/3 = 0
y2 - (4/3)y = - 1/3
(y - 2/3)2 = (2/3)2 - 1/3 = 4/9 - 1/3 = 1/9
Taking square roots: y - 2/3 = +/- 1/3
and so y = 2/3 +/- 1/3 => y = 1 or y = 1/3
Then x = y + 2 => x = 3 or x = 21/3.
So the points are (1, 3) and (21/3 , 1/3)
Rearrange the first equation to y = 2-x and then substitute this into the second equation to form the quadratic equation: -3y2+4y-1 = 0 and when solved y = 3 or y = 1/3 Points of intersection are: (3,1) and (2 and 1/3, 1/3)
If the circle is inside the square, four.
8
x-y = 0 => x = y x2+y2 = 18 => y2+y2 = 18 2y2 =18 Divide both sides by 2 and the square root both side: y = 3 Therfore the points of intersection are: (3, 3) and (-3, -3)
If you mean: 11x2-34x+3 = 0 then the solutions are x = 1/11 and x = 3 by completing the square or using the quadratic equation formula
Divide all terms by 3 so:- x2-4x = 5 Completing the square:- (x-2)2 = 9 x-2 = -/+3 x = -1 or x = 5
If: y = x+1 then y^2 = (x+1)^2 If: x^2 +y^2 = 25 then x^2 +(x+1)^2 -25 = 0 Multiplying out the brackets: x^2 +x^2 +2x +1 -25 = 0 Collecting like terms: 2x^2 +2x -24 = 0 Dividing all terms by 2: x^2 +x -12 = 0 Completing the square: (x+0.5)^2 = 12.25 Square root both sides: x+0.5 = -3.5 or +3.5 Deducting 0.5 from both sides: x = -4 or 3 Intersections by substitution into original equations are at: (-4,-3) and (3, 4)
x = 2 or x = 2/5
This quadratic equation has no solutions because the discriminant is less than zero.
The point(s) of intersection must satisfy both equations.So2x + k = x2 - 8x - 7Or, x2 - 10x - 7 - k = 0The discriminant for this quadratic equation is (-10)2 - 4*1*(-7-k)= 100 + 28 + 4k= 128 + 4kFor there to be 2 real distinct points of intersection, discriminant > 0that is 128 + 4k > 04k > -128 so that k > -32.For there to be a single point of intersection, discriminant = 0 so that k = -32.
Equations: x2+2x-7 = 17-3x Quadratic equation: x2+5x-24 = 0 Points of intersection: (-8, 41) and (3, 8) Length of line: (-8-3)2+(41-8)2 = 1210 and the square root of this is the length of the line which is about 34.78505426 or to be exact it is 11 times the square root of 10.
Completing the square would be the same as "Finding the square root" So an example would be 16. 16 is a perfect square so it would reduce to 4.