Let f(x) = 3x + 1 and g(x) = 2 - x
determine values for a so that
a) (a)
b) f(a^2) = g(2a)
so plug in your arguments.
f(a^2) = 3a^2+1
g(2a) = 2-2a
So now just solve
3a^2+1 = 2-2a
a = -1, 1/3
a) To determine the values for a so that (a), we need to set the two equations equal to each other and solve for a:
3a + 1 = 2 - a
Adding a to both sides:
4a + 1 = 2
Subtracting 1 from both sides:
4a = 1
Dividing both sides by 4:
a = 1/4
Therefore, the value for a so that (a) is a = 1/4.
b) To determine the values for a so that f(a^2) = g(2a), we need to substitute the values into the equations:
f(a^2) = g(2a)
Substituting f(x) and g(x):
3(a^2) + 1 = 2 - (2a)
Expanding and simplifying:
3a^2 + 1 = 2 - 2a
Rearranging:
3a^2 + 2a + 1 = 2
Subtracting 2 from both sides:
3a^2 + 2a - 1 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, let's use the quadratic formula:
a = (-b ± √(b^2 - 4ac)) / 2a
a = (-(2) ± √((2)^2 - 4(3)(-1))) / 2(3)
Calculating inside the square root:
a = (-2 ± √(4 + 12)) / 6
a = (-2 ± √(16)) / 6
Taking the square root:
a = (-2 ± 4) / 6
Simplifying:
a = (2, -1/3)
Therefore, the values for a so that f(a^2) = g(2a) are a = 2 and a = -1/3.
To determine the values of "a" that satisfy the given conditions, let's solve each equation step-by-step:
a) (a)
Step 1: Substitute the expressions for f(x) and g(x) into the equation.
3a + 1 = 2 - a
Step 2: Simplify the equation by combining like terms.
3a + a = 2 - 1
Step 3: Combine the terms on the left side.
4a = 1
Step 4: Isolate the variable "a" by dividing both sides by 4.
a = 1/4
Therefore, the value of "a" that satisfies the equation (a) is a = 1/4.
b) f(a^2) = g(2a)
Step 1: Substitute the expressions for f(x) and g(x) into the equation.
3(a^2) + 1 = 2 - (2a)
Step 2: Simplify the equation by expanding and combining like terms.
3a^2 + 1 = 2 - 2a
Step 3: Rearrange the equation to make it a quadratic equation in standard form.
3a^2 + 2a - 1 = 0
Step 4: Solve the quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, we will use the quadratic formula.
The quadratic formula is given by: a = (-b ± √(b^2 - 4ac)) / (2a)
For the equation 3a^2 + 2a - 1 = 0, the coefficients are:
a = 3, b = 2, c = -1
Applying the quadratic formula:
a = (-2 ± √((2^2) - 4(3)(-1))) / (2(3))
Simplifying:
a = (-2 ± √(4 + 12)) / 6
a = (-2 ± √16) / 6
a = (-2 ± 4) / 6
There are two solutions:
a1 = (-2 + 4) / 6 = 2 / 6 = 1/3
a2 = (-2 - 4) / 6 = -6 / 6 = -1
Therefore, the values of "a" that satisfy the equation f(a^2) = g(2a) are a = 1/3 and a = -1.
To determine the values of 'a' that satisfy the given conditions, let's solve each equation separately.
a) For (a):
Start by substituting f(x) and g(x) into the equation:
3a + 1 = 2 - a
Combine like terms:
4a + 1 = 2
Subtract 1 from both sides:
4a = 1
Divide both sides by 4:
a = 1/4
Therefore, the value of 'a' that satisfies (a) is a = 1/4.
b) For f(a^2) = g(2a):
Start by substituting f(x) and g(x) into the equation:
3(a^2) + 1 = 2 - (2a)
Simplify the equation:
3a^2 + 1 = 2 - 2a
Combine like terms:
3a^2 + 2a + 1 = 2
Rearrange the equation to set it equal to zero:
3a^2 + 2a - 1 = 0
To solve this quadratic equation, you can use the quadratic formula:
a = (-b ± √(b^2 - 4ac)) / (2a)
Considering a the coefficient of 'a^2', b the coefficient of 'a', and c the constant term, the equation becomes:
a = (-2 ± √(2^2 - 4 * 3 * -1)) / (2 * 3)
Simplify the equation:
a = (-2 ± √(4 + 12)) / 6
a = (-2 ± √16) / 6
a = (-2 ± 4) / 6
There are two possible solutions:
For the positive solution:
a = (-2 + 4) / 6 = 2 / 6 = 1 / 3
For the negative solution:
a = (-2 - 4) / 6 = -6 / 6 = -1
Therefore, the values of 'a' that satisfy f(a^2) = g(2a) are a = 1/3 and a = -1.