6x^2+21x-45
45 = 5*9
(1 5)(6 9) gives 30,9 gives 21
(x+5)(6x-9)
To factor the quadratic expression 6x^2 + 21x - 45, follow these steps:
Step 1: Look for any common factors among the coefficients of the quadratic terms. In this case, the coefficients 6, 21, and -45 have a common factor of 3. So, factor out 3 from the expression:
3(2x^2 + 7x - 15)
Step 2: Now we need to find two numbers whose product is -30 (the product of 2 and -15), and whose sum is 7 (the coefficient of the linear term 7x). These numbers are 10 and -3.
Step 3: Rewrite the linear term (7x) as the sum of -3x and 10x:
3(2x^2 - 3x + 10x - 15)
Step 4: Group the terms together with the common factors:
3((2x^2 - 3x) + (10x - 15))
Step 5: Factor out the greatest common factor from each group:
3(x(2x - 3) + 5(2x - 3))
Step 6: Notice that there is a common binomial factor of (2x - 3) in both terms. Factor it out:
3(2x - 3)(x + 5)
Therefore, the factored form of 6x^2 + 21x - 45 is 3(2x - 3)(x + 5).
The expression you provided is a quadratic polynomial: 6x^2 + 21x - 45.
To understand and solve this, we can use a few different methods such as factoring, completing the square, or using the quadratic formula. Let's go through each approach:
1. Factoring:
We try to factor the quadratic expression into two binomials. In this case, it may be a bit challenging as the coefficient of x^2 is 6, which is not easily factored. However, we can try to split the middle term.
The expression: 6x^2 + 21x - 45
First, we find the factors of the coefficient of x^2 (6) and the constant term (-45). The factors of 6 are 1, 2, 3, and 6, while the factors of 45 are 1, 3, 5, 9, 15, and 45.
Since the middle term is positive (21x), we look for the pair of factors that add up to 21. From the above list, the pair that satisfies this condition is 9 and 5 (9 + 5 = 14).
Now, we can rewrite the original expression by splitting the middle term 21x into 9x + 15x:
6x^2 + 9x + 15x - 45
Group the terms and factor by grouping:
(6x^2 + 9x) + (15x - 45)
3x(2x + 3) + 15(2x + 3)
Notice that we have a common binomial factor of (2x + 3):
(3x + 15)(2x + 3)
Now, we have factored the quadratic expression into two binomials: (3x + 15)(2x + 3). To find the roots (solutions), we set each factor equal to zero and solve for x.
Setting (3x + 15) equal to zero:
3x + 15 = 0
3x = -15
x = -5
Setting (2x + 3) equal to zero:
2x + 3 = 0
2x = -3
x = -3/2 or -1.5
Therefore, the solutions to the equation 6x^2 + 21x - 45 = 0 are x = -5, x = -3/2, or x = -1.5.
2. Completing the Square:
We can also solve this quadratic equation by completing the square.
The quadratic expression: 6x^2 + 21x - 45
Step 1: Divide the equation by the coefficient of x^2 (6) to make the coefficient of x^2 equal to 1:
x^2 + (21/6)x - (45/6)
Simplifying, we get x^2 + (7/2)x - 15/2
Step 2: Take half of the coefficient of x (7/2 in this case), square it, and add/subtract it from the equation. We add the square term inside the parentheses and subtract it outside to maintain the equality:
x^2 + (7/2)x + (7/4)^2 - (7/4)^2 - 15/2
Step 3: Rewrite the equation using the perfect square trinomial (a^2 + 2ab + b^2):
(x + 7/4)^2 - (49/16) - 15/2
Step 4: Simplify:
(x + 7/4)^2 - (49/16) - (120/16)
(x + 7/4)^2 - (169/16)
Now, the equation is in vertex form, (x - h)^2 - k, where (h, k) represents the coordinates of the vertex.
The vertex is at (-7/4, 169/16), which is obtained by changing the signs of h and k.
3. Quadratic Formula:
Finally, we can solve the quadratic equation using the quadratic formula:
For the equation: 6x^2 + 21x - 45 = 0
The quadratic formula states that for an equation in the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = 6, b = 21, and c = -45. Substituting these values into the formula:
x = (-(21) ± √((21)^2 - 4(6)(-45))) / 2(6)
Simplifying,
x = (-21 ± √(441 + 1080)) / 12
x = (-21 ± √1521) / 12
Taking the square root,
x = (-21 ± 39) / 12
Thus, the solutions are:
x = (-21 + 39) / 12 = 18 / 12 = 3/2 or 1.5
x = (-21 - 39) / 12 = -60 / 12 = -5
Hence, the solutions to the quadratic equation 6x^2 + 21x - 45 = 0 are x = -5, x = -3/2, or x = 1.5.