solve the quadratic equation by completing the square.
4x^2-3x-5=0
To solve the quadratic equation 4x^2 - 3x - 5 = 0 by completing the square, follow these steps:
Step 1: Ensure the coefficient of x^2 is 1. In this case, it is already 4, so no adjustment is needed.
Step 2: Move the constant term (in this case, -5) to the other side of the equation:
4x^2 - 3x = 5
Step 3: Divide both sides of the equation by the coefficient of x^2:
x^2 - (3/4)x = 5/4
Step 4: Take half of the coefficient of x (in this case, (3/4)/2 = 3/8), square it ((3/8)^2 = 9/64), and add it to both sides of the equation:
x^2 - (3/4)x + 9/64 = 5/4 + 9/64
Step 5: Simplify both sides:
x^2 - (3/4)x + 9/64 = 80/64 + 9/64
x^2 - (3/4)x + 9/64 = 89/64
Step 6: Factor the left side of the equation (the perfect square trinomial):
(x - 3/8)^2 = 89/64
Step 7: Take the square root of both sides:
x - 3/8 = ± √(89/64)
Step 8: Solve for x:
x = 3/8 ± √(89/64)
Hence, the solutions to the quadratic equation 4x^2 - 3x - 5 = 0, obtained by completing the square, are:
x = 3/8 + √(89/64) and x = 3/8 - √(89/64)
Step 1: Set the equation equal to zero:
4x^2 - 3x - 5 = 0
Step 2: Divide each term by the coefficient of the x^2 term:
x^2 - (3/4)x - (5/4) = 0
Step 3: Move the constant term to the right side:
x^2 - (3/4)x = (5/4)
Step 4: Take half of the coefficient of the x term, square it, and add it to both sides of the equation:
x^2 - (3/4)x + (-3/8)^2 = (5/4) + (-3/8)^2
x^2 - (3/4)x + 9/64 = 5/4 + 9/64
Step 5: Simplify the right side:
x^2 - (3/4)x + 9/64 = 80/64 + 9/64
x^2 - (3/4)x + 9/64 = 89/64
Step 6: Combine the terms on the left side:
(x - 3/8)^2 = 89/64
Step 7: Take the square root of both sides (note: we need to consider both positive and negative square roots):
x - 3/8 = ± √(89/64)
Step 8: Solve for x by adding 3/8 to both sides:
x = 3/8 ± √(89/64)
Therefore, the solutions to the quadratic equation 4x^2 - 3x - 5 = 0 by completing the square are:
x = 3/8 + √(89/64)
x = 3/8 - √(89/64)
To solve the quadratic equation 4x^2 - 3x - 5 = 0 by completing the square, follow these steps:
Step 1: Ensure that the quadratic equation is in the standard form, ax^2 + bx + c = 0. If necessary, rearrange the equation to put it in this form.
Given equation: 4x^2 - 3x - 5 = 0
Step 2: Divide the entire equation by the coefficient of x^2 to make the coefficient of x^2 equal to 1.
Divide the equation by 4:
(4x^2 - 3x - 5)/4 = 0
x^2 - (3/4)x - (5/4) = 0
Step 3: Move the constant term (the term without x) to the right side of the equation.
x^2 - (3/4)x = (5/4)
Step 4: To complete the square, take half of the coefficient of x, square it, and add it to both sides of the equation.
For our equation, the coefficient of x is -(3/4). So, half of -(3/4) is -(3/8), and when squared, it is (9/64).
Add (9/64) to both sides:
x^2 - (3/4)x + (9/64) = (5/4) + (9/64)
x^2 - (3/4)x + (9/64) = (80/64) + (9/64)
x^2 - (3/4)x + (9/64) = (89/64)
Step 5: Factor the perfect square trinomial on the left side of the equation.
The left side of the equation can be factored into a perfect square trinomial: (x - (3/8))^2
(x - (3/8))^2 = (89/64)
Step 6: Take the square root of both sides of the equation.
Taking the square root of both sides gives us:
x - (3/8) = ±( √(89/64) )
Step 7: Solve for x.
To isolate x, add (3/8) to both sides:
x = (3/8) ± ( √(89/64) )
Therefore, the solutions to the quadratic equation 4x^2 - 3x - 5 = 0 are:
x = (3/8) + ( √(89/64) )
x = (3/8) - ( √(89/64) )