Solve by completing the square
12x^2+11x=5
12(x^2+11/12 x + zzzz)=5+ zzzzz
well, to find zzzz, take 1/2*11/12, and square it.
121/24^2 check that.
12(x+11/24)^2=5+ (11/24)^2
(x+11/24)=+- sqrt ((1/12)(5+(11/24)^2)
so hold on what is the answer?
To solve the quadratic equation 12x^2 + 11x = 5 by completing the square, follow these steps:
Step 1: Move the constant term (5) to the other side of the equation.
12x^2 + 11x - 5 = 0
Step 2: Divide the entire equation by the coefficient of x^2 (12) to make the coefficient 1.
x^2 + (11/12)x - 5/12 = 0
Step 3: Take half of the coefficient of x and square it. Add this value to both sides of the equation.
The coefficient of x is (11/12). Half of it is (11/12) * (1/2) = 11/24. The square of this value is (11/24)^2 = 121/576.
x^2 + (11/12)x + 121/576 - 5/12 = 121/576
Simplifying the equation:
x^2 + (11/12)x + 121/576 - 240/576 = 0
x^2 + (11/12)x + (-119/576) = 0
Step 4: Factor the left side of the equation, if possible.
Unfortunately, the left side of the equation does not factor easily.
Step 5: Rewrite the equation as a perfect square trinomial.
Take half of the coefficient of x and square it: (11/12)^2 = 121/144.
Add this value to both sides of the equation.
x^2 + (11/12)x + 121/144 = 121/144 + (-119/576)
Simplifying the equation:
x^2 + (11/12)x + 121/144 = 169/576
Step 6: Rewrite the perfect square trinomial as a binomial squared.
The square root of the perfect square term (121/144) is 11/12.
(x + 11/12)^2 = 169/576
Step 7: Take the square root of both sides of the equation.
x + 11/12 = ±√(169/576)
Step 8: Simplify the square root expression.
√(169/576) = √169 / √576 = 13/24
Step 9: Solve for x
x + 11/12 = 13/24 or x + 11/12 = -13/24
For the first case:
x = 13/24 - 11/12 = 13/24 - 22/24 = -9/24 = -3/8
For the second case:
x = -13/24 - 11/12 = -13/24 - 22/24 = -35/24
Therefore, the solutions to the equation 12x^2 + 11x = 5 are x = -3/8 and x = -35/24.