Solve the quadratic equation by completing the square.
-3x^2+9x=1
Not sure how to do these, if someone could expplain.
Still confused, but I'll try another one and see if I get it. Thank you.
To solve the quadratic equation -3x^2 + 9x = 1 by completing the square, follow these steps:
Step 1: Set the equation equal to zero:
-3x^2 + 9x - 1 = 0
Step 2: Divide the entire equation by the coefficient of x^2:
-3(x^2 - 3x/(-3) + (-1)/(-3)) = 0
This simplifies to:
-3(x^2 - 3x + 1/3) = 0
Step 3: Complete the square by adding the square of half the coefficient of x to both sides:
To find the square of half the coefficient of x:
(-3/2 * (-3/2)) = 9/4
Add this value to both sides of the equation:
-3(x^2 - 3x + 1/3) + 9/4 = 9/4
Simplify:
-3(x^2 - 3x + 1/3 + 3/4) = 9/4
Step 4: Factor the perfect square trinomial:
-3[(x - 3/2)^2 + 1/3 + 3/4] = 9/4
Simplify the expression inside the square brackets:
(x - 3/2)^2 + 1/9 + 1/4 = -3/4
Step 5: Combine the constant terms on the right-hand side:
(x - 3/2)^2 + (4/36) + (9/36) = -3/4
Simplify:
(x - 3/2)^2 + 13/36 = -3/4
Step 6: Move the constant term (13/36) to the right-hand side:
(x - 3/2)^2 = -3/4 - 13/36
Simplify the right-hand side:
(x - 3/2)^2 = -9/12 - 13/36
Step 7: Find the common denominator and combine the fractions on the right-hand side:
(x - 3/2)^2 = (-27 - 13)/36
Simplify the numerator:
(x - 3/2)^2 = -40/36
Step 8: Simplify the right-hand side:
(x - 3/2)^2 = -10/9
Step 9: Take the square root of both sides to eliminate the square:
x - 3/2 = ±√(-10/9)
Step 10: Simplify the right-hand side:
x - 3/2 = ±√(-10)/√9
Since the square root of a negative number is not a real number, the equation has no real solutions.
To solve a quadratic equation by completing the square, follow these steps:
Step 1: Make sure the quadratic equation is in the standard form, which is ax^2 + bx + c = 0. In your case, the equation is already in this form: -3x^2 + 9x = 1.
Step 2: Divide the entire equation by the coefficient of x^2 to make the leading coefficient 1. In this case, divide both sides of the equation by -3:
(-3x^2 + 9x)/-3 = 1/-3
Simplifying gives: x^2 - 3x = -1/3.
Step 3: Move the constant term to the other side of the equation. In this case, add 1/3 to both sides:
x^2 - 3x + 1/3 = 0.
Step 4: Complete the square of the quadratic expression on the left side of the equation. To do this, take half of the coefficient of x (-3/2), square it, and add it to both sides:
x^2 - 3x + 9/4 = 9/4 + 1/3
Combining the fractions on the right side gives: x^2 - 3x + 9/4 = 27/12 + 4/12 = 31/12.
Step 5: Rewrite the left side of the equation as a perfect square and simplify the right side. The left side is a perfect square since (x - 3/2)^2 = x^2 - 3x + 9/4:
(x - 3/2)^2 = 31/12.
Step 6: Take the square root of both sides of the equation. Remember to consider both the positive and negative square roots:
x - 3/2 = ±√(31/12).
Step 7: Isolate x by adding 3/2 to both sides of the equation:
x = 3/2 ±√(31/12).
So, the solutions to the quadratic equation -3x^2 + 9x = 1, obtained by completing the square, are x = 3/2 + √(31/12) and x = 3/2 - √(31/12).
it's usually easier to have a positive coefficient for x^2, so let's take care of that first:
3x^2 - 9x = -1
x^2 - 3x = -1/3
since (x-a)^2 = x^2 - 2a + a^2, you want to take half of 3 and square it, then add that to both sides:
x^2 - 3x + 9/4 = -1/3 + 9/4
Now you have a perfect square on the left:
(x-3/2)^2 = 23/12
now take square roots:
x - 3/2 = ±√(23/12) = ±√69/6
so,
x = 3/2 ±√69/6
or,
x = (9±√69)/6
Note how that is the solution using the quadratic formula.