Solve the quadratic equation by completing the square.
3x^2-5x-4=0
The first step in solving this equation by completing the square is to write the equation in the form of a perfect square trinomial.
Starting from the equation:
3x^2 - 5x - 4 = 0
You then isolate the x-terms on one side of the equation:
3x^2 - 5x = 4
Then, divide all terms by '3' to ensure that the coefficient (number associated with) of x^2 equals 1:
x^2 - (5/3)x = 4/3
Next, you can take half of the coefficient of x (which is -5/3), square it, and then add and subtract that number, on the left side of the equation (because the right side is considered as 0, so 4/3 - 4/3 = 0):
x^2 - (5/3)x + (5/6)^2 = 4/3 + (5/6)^2
It simplifies into:
(x - 5/6)^2 = 4/3 + 25/36
Perform the addition on the right side of the equation:
(x - 5/6)^2 = 77/36
The final step is to take the square root of both sides
x - 5/6 = √(77/36)
To get x alone, you then add 5/6 to both sides of the equation:
x = √(77/36) + 5/6
So the solutions of the quadratic equation are
x = √(77/36) + 5/6 and x = -√(77/36) + 5/6:
x = √(77/36) + 5/6 ≈ 1.89
x = -√(77/36) + 5/6 ≈ -0.22.
Step 1: Make sure the quadratic equation is in the form ax^2 + bx + c = 0. In this case, the equation is already in this form.
3x^2 - 5x - 4 = 0
Step 2: Divide all terms by the coefficient of x^2 if it is not already 1 to simplify the equation. In this case, the coefficient of x^2 is already 3, so no division is needed.
Step 3: Move the constant term (c) to the right side of the equation.
3x^2 - 5x = 4
Step 4: Add the square of half of the coefficient of x to both sides of the equation. The coefficient of x is -5, so half of it is -5/2. The square of -5/2 is 25/4.
3x^2 - 5x + 25/4 = 4 + 25/4
Simplifying the right side:
4 + 25/4 = 16/4 + 25/4 = 41/4
The equation becomes:
3x^2 - 5x + 25/4 = 41/4
Step 5: Rewrite the left side as a perfect square trinomial.
3(x^2 - (5/3)x) + 25/4 = 41/4
To complete the square, take half of the coefficient of x (-5/3), square it, and add it to both sides of the equation.
3(x^2 - (5/3)x + (-5/6)^2) + 25/4 = 41/4 + 25/4
Simplifying the right side:
41/4 + 25/4 = 66/4 = 33/2
The equation becomes:
3(x - 5/6)^2 + 25/4 = 33/2
Step 6: Move the constant term (25/4) to the right side of the equation.
3(x - 5/6)^2 = 33/2 - 25/4
Simplifying the right side:
33/2 - 25/4 = 66/4 - 25/4 = 41/4
The equation becomes:
3(x - 5/6)^2 = 41/4
Step 7: Divide both sides of the equation by 3 to isolate the perfect square expression.
(x - 5/6)^2 = 41/4 / 3
Simplifying the right side:
41/4 / 3 = 41/4 * 1/3 = 41/12
The equation becomes:
(x - 5/6)^2 = 41/12
Step 8: Take the square root of both sides of the equation.
√((x - 5/6)^2) = ±√(41/12)
Simplifying the left side:
√((x - 5/6)^2) = |x - 5/6|
Simplifying the right side:
±√(41/12) = ±(√41) / (√12)
Step 9: Solve for x by adding 5/6 to both sides of the equation.
x - 5/6 = ±(√41) / (√12)
To isolate x, add 5/6 to both sides:
x = 5/6 ± (√41) / (√12)
This is the solution to the quadratic equation by completing the square.
To solve the given quadratic equation by completing the square, follow these steps:
Step 1: Set the equation equal to zero.
3x^2 - 5x - 4 = 0
Step 2: Divide the entire equation by the coefficient of x^2 to make the coefficient 1.
(3x^2 - 5x - 4)/3 = 0
Step 3: Move the constant term to the other side of the equation.
3x^2 - 5x = 4
Step 4: Move the x term to the other side by dividing the coefficient by 2 and squaring it.
3x^2 - 5x + (5/6)^2 = 4 + (5/6)^2
Step 5: Simplify the equation.
3x^2 - 5x + 25/36 = 144/36 + 25/36
Step 6: Combine the fractions on the right side.
3x^2 - 5x + 25/36 = 169/36
Step 7: Rewrite the left side as a binomial squared.
(x - 5/6)^2 = 169/36
Step 8: Take the square root of both sides.
sqrt((x - 5/6)^2) = ±sqrt(169/36)
Step 9: Simplify the equation.
x - 5/6 = ±13/6
Step 10: Solve for x.
x = 5/6 ± 13/6
So the solutions to the quadratic equation 3x^2 - 5x - 4 = 0 are x = (5 + 13)/6 and x = (5 - 13)/6, which can be further simplified as x = 3 and x = -2/3.