Solve the quadratic by factoring.
3, x, squared, minus, 8, x, minus, 1, equals, minus, 6, x, plus, 4
3x
2
−8x−1=−6x+4
To begin, let's write the equation in standard quadratic form:
3x^2 - 8x - 1 = -6x + 4
Now, let's move all terms to one side to set the equation equal to zero:
3x^2 - 8x + 6x - 5 = 0
Combine like terms:
3x^2 - 2x - 5 = 0
We can now look for factors of -5 that add to -2. The factors are -5 and 1.
Split the middle term using these factors:
3x^2 - 5x + 3x - 5 = 0
Rearrange the terms:
(3x^2 - 5x) + (3x - 5) = 0
Factor out the greatest common factor from each group:
x(3x - 5) + (3x - 5) = 0
Now, we can see that both terms have a common factor (3x - 5). Combine like terms:
(3x - 5)(x + 1) = 0
To find the solutions, set each factor equal to zero:
3x - 5 = 0 --> x = 5/3
x + 1 = 0 --> x = -1
Therefore, the solutions to the quadratic equation are x = 5/3 and x = -1.
To solve the quadratic equation 3x^2 - 8x - 1 = -6x + 4 by factoring, we can follow these steps:
Step 1: Rewrite the equation in standard form:
3x^2 - 8x - 1 + 6x - 4 = 0
3x^2 - 2x - 5 = 0
Step 2: Factor the quadratic expression:
To factor the quadratic expression, we need to find two numbers that multiply to -15 (the coefficient of x^2 multiplied by the constant term) and add up to -2 (the coefficient of x).
Let's try to find those numbers by listing the factors of -15:
-15 and 1
-5 and 3
-3 and 5
-1 and 15
None of these pairs of numbers adds up to -2, so this quadratic cannot be factored using integer values.
Step 3: Use the quadratic formula:
If the quadratic cannot be factored, we can use the quadratic formula to find the solutions. The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation 3x^2 - 2x - 5 = 0, the values of a, b, and c are:
a = 3
b = -2
c = -5
Substituting these values into the quadratic formula, we have:
x = (-(-2) ± √((-2)^2 - 4(3)(-5))) / (2(3))
x = (2 ± √(4 + 60)) / 6
x = (2 ± √64) / 6
x = (2 ± 8) / 6
So, the two possible solutions are:
x = (2 + 8) / 6
x = 10 / 6
x = 5/3
and
x = (2 - 8) / 6
x = -6 / 6
x = -1
Therefore, the quadratic equation 3x^2 - 8x - 1 = -6x + 4 has two solutions:
x = 5/3 and x = -1.
To solve the quadratic equation 3x^2 - 8x - 1 = -6x + 4 by factoring, we will follow these steps:
Step 1: Rewrite the equation in the standard form with zero on one side: 3x^2 - 8x - 1 + 6x - 4 = 0
Step 2: Combine like terms: 3x^2 - 2x - 5 = 0
Step 3: We need to find two numbers that multiply to give -15 (product of the coefficient of x^2 term and the constant term) and add up to -2 (coefficient of the x term). The numbers -5 and 3 satisfy this condition.
Step 4: Rewrite the middle term -2x as the sum of -5x and 3x: 3x^2 - 5x + 3x - 5 = 0
Step 5: Group the terms and factor by grouping:
(x^2 - 5x) + (3x - 5) = 0
Step 6: Factor out the greatest common factor from each group:
x(x - 5) + 1(3x - 5) = 0
Step 7: Factor out the binomial factors:
(x - 5)(x + 1) = 0
Step 8: Set each factor equal to zero and solve for x:
x - 5 = 0 or x + 1 = 0
Step 9: Solve for x:
x = 5 or x = -1
Therefore, the solutions to the quadratic equation 3x^2 - 8x - 1 = -6x + 4 are x = 5 and x = -1.