topic: quadratic equation.
used completing the square methods
7v²+22v+3
7v^2 + 22v + 3
7(v^2 + 22/7 v) + 3
7(v^2 + 22/7 v + (11/7)^2) + 3 - 7(11/7)^2
7(v + 11/7)^2 - 100/7
To solve the quadratic equation 7v² + 22v + 3 using the completing the square method, follow these steps:
Step 1: Set the equation equal to zero
7v² + 22v + 3 = 0
Step 2: Make sure the coefficient of the quadratic term (v²) is 1. In this case, it is already 7v², so no changes are needed.
Step 3: Divide the coefficient of the linear term (v) by 2, square the result, and add it to both sides of the equation:
Coefficient of the linear term (v) / 2 = 22/2 = 11
Square of 11 = 121
Add 121 to both sides of the equation:
7v² + 22v + 121 + 3 = 121
Step 4: Rewrite the left side of the equation as a perfect square trinomial:
7v² + 22v + 124 = 121
Step 5: Factor the perfect square trinomial. To do this, take the square root of the first term, double it, and place it in the middle term:
(√7v + 11)² = 121
Step 6: Take the square root of both sides to solve for v:
√[(√7v + 11)²] = ±√121
√7v + 11 = ±11
Step 7: Solve for v by isolating it on one side of the equation:
√7v = ±11 - 11
√7v = ±0
Step 8: Square both sides of the equation to solve for v:
7v = 0
Step 9: Divide both sides by 7 to get the final solution for v:
v = 0
So, the solutions to the quadratic equation 7v² + 22v + 3 = 0 using the completing the square method are v = 0.
To solve the quadratic equation 7v² + 22v + 3 using the completing the square method, follow these steps:
Step 1: Identify the coefficients a, b, and c from the quadratic equation. In this case, a = 7, b = 22, and c = 3.
Step 2: Rewrite the equation in the form (av² + bv + c) = 0.
7v² + 22v + 3 = 0
Step 3: Divide the entire equation by the coefficient of v² to make the coefficient of v² equal to 1. In this case, divide the equation by 7.
(7v² + 22v + 3)/7 = 0
Simplifying the equation, we get:
v² + (22/7)v + 3/7 = 0
Step 4: Move the constant term (c) to the other side of the equation.
v² + (22/7)v = -3/7
Step 5: Take half of the coefficient of v, square it, and add it to both sides of the equation.
v² + (22/7)v + (22/14)² = -3/7 + (22/14)²
Simplifying, we get:
v² + (22/7)v + (11/7)² = -3/7 + (11/7)²
Step 6: Simplify the right-hand side of the equation and rewrite the left-hand side as a perfect square.
v² + (22/7)v + (11/7)² = -3/7 + 121/49
v² + (22/7)v + (11/7)² = (121 - 21)/49
v² + (22/7)v + (11/7)² = 100/49
Step 7: Simplify the equation and write it as a perfect square on the left-hand side.
(v + 11/7)² = (10/7)²
Step 8: Take the square root of both sides of the equation.
v + 11/7 = ±(10/7)
Step 9: Solve for v.
v = -11/7 ± 10/7
So the solutions to the quadratic equation 7v² + 22v + 3 = 0 using the completing the square method are:
v = (-11/7) + (10/7) = -1/7
v = (-11/7) - (10/7) = -3