Solve the equation by completing the square: 4x^2 – 16x – 5 = 0
To solve this equation by completing the square, we need to rewrite the equation in the form of (x - h)^2 = k.
First, let's move the constant term to the other side of the equation:
4x^2 - 16x = 5
Next, divide the entire equation by the coefficient of x^2, which is 4, to make the coefficient of x^2 equal to 1:
x^2 - 4x = 5/4
Now, we need to complete the square by adding and subtracting a constant term to the equation. Take half of the coefficient of x (-4) and square it:
((-4)/2)^2 = (-2)^2 = 4
Add 4 to both sides of the equation:
x^2 - 4x + 4 = 5/4 + 4
We have now completed the square on the left side of the equation. Simplify the right side:
x^2 - 4x + 4 = 5/4 + 16/4
x^2 - 4x + 4 = 21/4
Now, factor the left side of the equation:
(x - 2)^2 = 21/4
Take the square root of both sides to solve for x:
x - 2 = ±√(21/4)
x - 2 = ±√21/2
Next, isolate x by adding 2 to both sides of the equation:
x = 2 ± √21/2
Therefore, the solutions to the equation 4x^2 – 16x – 5 = 0 are:
x = 2 + √21/2 and x = 2 - √21/2