Solve the equation by completing the square. If the solutions are real, give exact and approximate answers. Otherwise, list the exact solutions. 7x^2-4x+1=2x^2-7x+3 Please help!!!!! :(
To solve the given equation by completing the square, follow these steps:
Step 1: Move all terms to one side of the equation to set it equal to zero:
7x^2 - 4x + 1 - (2x^2 - 7x + 3) = 0
Simplifying the equation gives:
5x^2 + 3x - 2 = 0
Step 2: Divide the entire equation by the coefficient of x^2 to make the leading coefficient 1:
(x^2 + (3/5)x - (2/5)) = 0
Step 3: Now, focus on the quadratic term, x^2 + (3/5)x, and complete the square by adding the square of half the coefficient of x to both sides of the equation.
The coefficient of x is (3/5), so half of that is (3/10), and its square is (9/100):
(x^2 + (3/5)x + (9/100)) - (9/100) - (2/5) = 0
Simplifying the equation further gives:
(x + (3/10))^2 - (19/100) = 0
Step 4: Simplify and rewrite the equation:
(x + (3/10))^2 = 19/100
Step 5: Take the square root of both sides of the equation:
√((x + (3/10))^2) = ±√(19/100)
Simplifying gives:
x + (3/10) = ±√19/10
Step 6: Solve for x:
x = -3/10 ± √19/10
Therefore, the exact solutions are:
x = (-3 ± √19)/10
To find approximate solutions, you can use a calculator:
x ≈ -0.027 and x ≈ 1.227
So, the solutions to the equation 7x^2 - 4x + 1 = 2x^2 - 7x + 3 are x ≈ -0.027 and x ≈ 1.227.
To solve the equation 7x^2 - 4x + 1 = 2x^2 - 7x + 3 by completing the square, follow these steps:
Step 1: Move all the terms to one side to set the equation equal to zero:
7x^2 - 2x^2 - 4x + 7x + 1 - 3 = 0
5x^2 - 3x - 2 = 0
Step 2: Divide the entire equation by the coefficient of x^2 to make the leading coefficient equal to 1:
x^2 - (3/5)x - (2/5) = 0
Step 3: To complete the square, the coefficient of x must be multiplied by 1/2 and squared. Add the result inside the parentheses and subtract it outside the parentheses to keep the equation balanced:
x^2 - (3/5)x + (9/100) - (9/100) - (2/5) = 0
Step 4: Simplify the equation:
(x - 3/10)^2 - 9/100 - 20/100 = 0
(x - 3/10)^2 - 29/100 = 0
Step 5: Move the constant term to the other side:
(x - 3/10)^2 = 29/100
Step 6: Take the square root of both sides:
x - 3/10 = ± √(29/100)
x - 3/10 = ± (1/10)√29
Step 7: Solve for x:
x = 3/10 ± (1/10)√29
The solutions to the equation are:
x = 3/10 + (1/10)√29
x = 3/10 - (1/10)√29
These are the exact solutions. To obtain the approximate solutions, you can substitute the value of √29 with its decimal approximation and perform the necessary calculations.
No problem. First, collect things into a useful form.
7x^2-4x+1=2x^2-7x+3
5x^2+3x=2
5(x^2+3/5 x) = 2
Now, divide 3/5 by 2 and square it.
Make sure to add the same amount to both sides.
5(x^2 + 3/5 x + (3/10)^2) = 2 + 5(3/10)^2
5(x + 3/10)^2 = 245/100
(x + 3/10)^2 = 45/100
x + 3/10 = ±3√5/10
x = (-3±3√5)/10