Solve the quadratic equation 84x^2+16x=0
by factoring out the GCF.
To solve the quadratic equation 84x^2 + 16x = 0 by factoring out the greatest common factor (GCF), first, let's find the GCF of the coefficients 84 and 16.
The prime factorization of 84 is 2^2 * 3 * 7, and the prime factorization of 16 is 2^4.
The common factors are 2^2 = 4.
Now, let's factor out the GCF from the equation:
4(21x^2 + 4x) = 0
Now, we have a quadratic equation of the form ax^2 + bx = 0, where a = 21 and b = 4.
To solve this equation, we can set each factor equal to zero:
4 = 0 (This cannot be true)
21x^2 + 4x = 0
By factoring the quadratic equation further, we can now solve for x:
x(21x + 4) = 0
Setting each factor equal to zero:
x = 0
21x + 4 = 0
Solving the linear equation:
21x = -4
x = -4/21
Therefore, the solutions to the quadratic equation 84x^2 + 16x = 0, factored by the GCF, are x = 0 and x = -4/21.
To solve the quadratic equation 84x^2 + 16x = 0 by factoring out the greatest common factor (GCF), you first need to find the GCF of the terms 84x^2 and 16x.
Step 1: Find the GCF of 84x^2 and 16x.
The GCF of 84x^2 and 16x is 4x, as it is the greatest common factor that divides both terms evenly.
Step 2: Factor out the GCF.
To factor out 4x from the equation, divide each term by 4x:
(84x^2 + 16x)/4x = 0
(4x * (21x + 4))/4x = 0
Step 3: Simplify the equation.
Cancel out the common factor of 4x from the numerator and the denominator:
21x + 4 = 0
Now, you have factored the equation by factoring out the GCF. The quadratic equation 84x^2 + 16x = 0 can be simplified to 21x + 4 = 0.
To solve the quadratic equation 84x^2 + 16x = 0 by factoring out the Greatest Common Factor (GCF), first, let's find the GCF of the two terms:
84x^2 has the factors 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84.
16x has the factors 1, 2, 4, 8, and 16.
The largest common factor is 4, so we can factor out 4 from both terms:
4(21x^2 + 4x) = 0
Now, we have factored out the GCF. To solve for x, we set each factor equal to zero:
4 = 0 or (21x^2 + 4x) = 0
Since 4 does not equal zero, we ignore it for now and focus on solving the equation (21x^2 + 4x) = 0.
To solve this quadratic equation, we can now factor it further or use the quadratic formula. In this case, factoring is a bit difficult, so let's use the quadratic formula.
The quadratic formula is given by:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
For our equation (21x^2 + 4x) = 0, we have:
a = 21
b = 4
c = 0
Plugging these values into the quadratic formula, we get:
x = (-4 ± sqrt(4^2 - 4 * 21 * 0)) / (2 * 21)
x = (-4 ± sqrt(16)) / 42
x = (-4 ± 4) / 42
Now, we have two possible values for x:
x₁ = (-4 + 4) / 42 = 0 / 42 = 0
x₂ = (-4 - 4) / 42 = -8 / 42 = -4/21
Therefore, the solutions to the quadratic equation 84x^2 + 16x = 0 are x = 0 and x = -4/21.