4. Solve the equation using all three methods below.
Factoring, completing the square, quadratic formula.
x^2 + 3x - 4 = 0
and
4x^2 = 9
it seems like x^2 + 3x - 4 = 0 equals x = 1, -4 from all three methods but Im not sure
4x^2 = 9 equals x = 3/2, -(3/2) when using quadratic formula or factoring, but i don't think it can be solved by completing the square.
not sure? did you try the values? Looks good to me, since
1+3-4=0
16-12-4=0
for 4x^2=9 you don't really need to complete the square, but if you insist,
4x^2-9 = 0
4x^2 = 9
(2x-0)^2 = 9
2x-0 = ±3
x = 0±3/2
To solve the equation x^2 + 3x - 4 = 0 using different methods, we can start with factoring:
Method 1: Factoring
1. Rewrite the equation in factored form: (x - 1)(x + 4) = 0
2. Set each factor equal to zero:
(x - 1) = 0 or (x + 4) = 0
3. Solve for x by adding the constant term:
x = 1 or x = -4
Method 2: Completing the Square
1. Rearrange the equation with the constant term on one side:
x^2 + 3x = 4
2. Take half of the coefficient of the x-term (3/2)^2 = 9/4, and add it to both sides:
x^2 + 3x + 9/4 = 4 + 9/4
x^2 + 3x + 9/4 = 25/4
3. Rewrite the left side as a perfect square:
(x + 3/2)^2 = 25/4
4. Take the square root of both sides and solve for x:
x + 3/2 = ±√(25/4)
x = -3/2 ± 5/2
x = -3/2 + 5/2 or x = -3/2 - 5/2
x = 1 or x = -4
Method 3: Quadratic Formula
1. Start with the equation in the form ax^2 + bx + c = 0. In this case, a = 1, b = 3, and c = -4.
2. Substitute the values into the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
x = (-(3) ± √((3)^2 - 4(1)(-4)))/(2(1))
x = (-3 ± √(9 + 16))/2
x = (-3 ± √(25))/2
x = (-3 ± 5)/2
x = -8/2 or x = 2/2
x = -4 or x = 1
For the equation 4x^2 = 9, we can solve it using the same methods:
Method 1: Factoring
1. Rearrange the equation: 4x^2 - 9 = 0
2. Recognize that this equation is in the form of a difference of squares: (2x - 3)(2x + 3) = 0
3. Set each factor equal to zero:
(2x - 3) = 0 or (2x + 3) = 0
4. Solve for x by adding the constant term and dividing by the coefficient:
x = 3/2 or x = -3/2
Method 3: Quadratic Formula
1. Start with the equation in the form ax^2 + bx + c = 0. In this case, a = 4, b = 0, and c = -9.
2. Substitute the values into the quadratic formula:
x = (-0 ± √((0)^2 - 4(4)(-9)))/(2(4))
x = (0 ± √(0 + 144))/8
x = (0 ± √(144))/8
x = (0 ± 12)/8
x = 12/8 or x = 0/8
x = 3/2 or x = 0
As you can see, for the given equations, all three methods yield the same solutions: x = 1 and x = -4 for the first equation, and x = 3/2 and x = -3/2 for the second equation. Completing the square method can be used to solve the second equation as well, but it involves more steps compared to factoring or using the quadratic formula.