solve the equation by completing the square.
1) x squared-8x-4=0
2)x squared+7x=-12
To solve an equation by completing the square, follow these steps:
1) Move the constant term (number without any variable) to the right side of the equation, keeping all the terms with variables on the left side, so that the equation is in the form "ax^2 + bx + c = 0."
1) x^2 - 8x - 4 = 0
Move the constant term (-4) to the right side:
x^2 - 8x = 4
2) Take half of the coefficient of the x-term (b/2), square it, and add it to both sides of the equation. This step is called completing the square.
a) x^2 - 8x = 4
b) Take half of -8: -8/2 = -4
c) Square -4: (-4)^2 = 16
d) Add 16 to both sides:
x^2 - 8x + 16 = 4 + 16
x^2 - 8x + 16 = 20
3) Rewrite the left side as a perfect square binomial.
a) x^2 - 8x + 16 = 20
b) The left side can be factored as a perfect square: (x - 4)^2
c) Rewrite the equation: (x - 4)^2 = 20
4) Take the square root of both sides and consider both the positive and negative square roots.
a) (x - 4)^2 = 20
b) Take the square root of both sides:
sqrt((x - 4)^2 )= sqrt(20)
c) Simplify:
x - 4 = +/- sqrt(20)
d) Further simplify:
x - 4 = +/- 2sqrt(5)
5) Solve for x by isolating it on one side of the equation.
a) x - 4 = 2sqrt(5) or x - 4 = -2sqrt(5)
b) Add 4 to both sides:
x = 4 + 2sqrt(5) or x = 4 - 2sqrt(5)
Therefore, the solution to the equation x^2 - 8x - 4 = 0 by completing the square is x = 4 + 2sqrt(5) or x = 4 - 2sqrt(5).
Now let's solve the second equation using the same method.
2) x^2 + 7x = -12
3) Move the constant term (-12) to the right side:
x^2 + 7x = -12
4) Complete the square:
a) x^2 + 7x = -12
b) Take half of 7: 7/2 = 3.5
c) Square 3.5: (3.5)^2 = 12.25
d) Add 12.25 to both sides:
x^2 + 7x + 12.25 = -12 + 12.25
x^2 + 7x + 12.25 = 0.25
5) Rewrite the left side as a perfect square binomial:
a) x^2 + 7x + 12.25 = 0.25
b) The left side can be factored as a perfect square: (x + 3.5)^2
c) Rewrite the equation:
(x + 3.5)^2 = 0.25
6) Take the square root of both sides and consider both the positive and negative square roots:
a) (x + 3.5)^2 = 0.25
b) Take the square root of both sides:
sqrt((x + 3.5)^2) = sqrt(0.25)
c) Simplify:
x + 3.5 = +/- sqrt(0.25)
d) Further simplify:
x + 3.5 = +/- 0.5
7) Solve for x by isolating it on one side of the equation:
a) x + 3.5 = 0.5 or x + 3.5 = -0.5
b) Subtract 3.5 from both sides:
x = 0.5 - 3.5 or x = -0.5 - 3.5
x = -3 or x = -4
Therefore, the solution to the equation x^2 + 7x = -12 by completing the square is x = -3 or x = -4.