solve by completing square method 16x^2-24x-13=0
16x^2-24x-13=0
16(x^2 - 3/2 x) = 13
16(x^2 - 3/2 x + (3/4)^2 = 13+16(3/4)^2
16(x-3/4)^2 = 22
(x-3/4)^2 = 11/8
x = 3/4 ±√(11/8)
or, x = (3±√22)/4
or, eschewing fractions,
16x^2-24x-13 = 0
16x^2-24x+9 = 13+9
(4x-3)^2 = 22
...
To solve the quadratic equation 16x^2 - 24x - 13 = 0 using the completing the square method, follow these steps:
Step 1: Move the constant term to the other side of the equation:
16x^2 - 24x = 13
Step 2: Divide the entire equation by the coefficient of x^2 to make the coefficient of x^2 equal to 1:
x^2 - (24/16)x = 13/16
Step 3: Take half of the coefficient of x and square it. Add the result to both sides of the equation:
x^2 - (24/16)x + (24/16)^2 = 13/16 + (24/16)^2
x^2 - (24/16)x + (3/4)^2 = 13/16 + (3/4)^2
x^2 - (24/16)x + 9/16 = 13/16 + 9/16
Step 4: Simplify the equation on both sides:
x^2 - (24/16)x + 9/16 = 22/16
x^2 - (3/4)x + 9/16 = 11/8
Step 5: Write the left side of the equation as a perfect square:
(x - 3/4)^2 = 11/8
Step 6: Take the square root of both sides of the equation:
√((x - 3/4)^2) = ±√(11/8)
Step 7: Simplify the square root on the left side and right side of the equation:
x - 3/4 = ±√(11/8)
Step 8: Add 3/4 to both sides of the equation:
x = 3/4 ±√(11/8)
Therefore, the solutions to the quadratic equation 16x^2 - 24x - 13 = 0, using the completing the square method, are:
x = 3/4 + √(11/8) and x = 3/4 - √(11/8)
To solve the equation 16x^2 - 24x - 13 = 0 using the completing the square method, follow these steps:
Step 1: Make sure the equation is in the standard form, ax^2 + bx + c = 0. In this case, it already is, so we can move on to the next step.
Step 2: Divide the equation through by the coefficient of x^2 (a = 16) to make the coefficient of x^2 equal to 1.
16x^2 - 24x - 13 = 0
Divide by 16:
x^2 - (24/16)x - (13/16) = 0
Simplifying, we get:
x^2 - (3/2)x - (13/16) = 0
Step 3: Move the constant term (C) to the other side of the equation.
x^2 - (3/2)x = (13/16)
Step 4: Take half of the coefficient of x (b) and square it [(b/2)^2]. Add this quantity to both sides of the equation.
x^2 - (3/2)x + [(3/4)^2] = (13/16) + [(3/4)^2]
(x - 3/4)^2 = (13/16) + 9/16
(x - 3/4)^2 = 22/16
Step 5: Simplify the equation.
(x - 3/4)^2 = 11/8
Step 6: Take the square root of both sides of the equation.
x - 3/4 = ±√(11/8)
Step 7: Solve for x.
x = 3/4 ±√(11/8)
These are the two solutions for the equation 16x^2 - 24x - 13 = 0 using the completing the square method.