Hey, I have another question. this I just don't understand. I'm really starting to get very confused!!! PLEASE HELP!!!!!
Solve each equation by factoring. Check your answers.
3x^2+45=24x
3x^2 - 24x + 45 = 0
remember Steve told you about all that "zero stuff" ?
divide each term by 3
x^2 - 8x + 15 = 0
can you think of 2 numbers that add up to -8 and multiply to get 15 ?
I am thinking -3 and -5
so
(x-3)(x-5) = 0
x-3=0 ----> x = 3
or
x-5 = 0 ----> x = 5
Sure, I can help you with that. To solve the equation 3x^2 + 45 = 24x by factoring, we need to rearrange the equation to bring all terms to one side equal to zero.
So let's start by subtracting 24x from both sides:
3x^2 - 24x + 45 = 0
Now we can attempt to factor the quadratic expression 3x^2 - 24x + 45. But before that, let's see if it can be factored by looking at the discriminant of the quadratic formula.
The discriminant (D) of a quadratic equation ax^2 + bx + c = 0 is given by D = b^2 - 4ac. If the discriminant is negative (D < 0), the quadratic equation cannot be factored nicely and we would need to use a different method such as the quadratic formula.
In this case, let's calculate the discriminant:
D = (-24)^2 - 4(3)(45)
D = 576 - 540
D = 36
Since the discriminant (D) is positive (D > 0), we can proceed with factoring.
Now, we need to find two numbers that multiply to give us the product of a * c (where a = 3, b = -24, and c = 45) and add up to give b (which is -24 in our case).
The product of a * c is 3 * 45 = 135, and we need two numbers that add up to -24.
Looking at the factors of 135, we can find the pair (9, 15) that satisfies the conditions:
9 + 15 = 24 and 9 * 15 = 135
Now, we can rewrite the middle term -24x as -9x - 15x:
3x^2 - 9x - 15x + 45 = 0
And then, we can group the terms and factor by grouping:
(3x^2 - 9x) + (-15x + 45) = 0
3x(x - 3) - 15(x - 3) = 0
Notice that we have a common factor of (x - 3). We can factor it out:
(x - 3)(3x - 15) = 0
Now, we have two factors that multiply to zero. So, we can set each factor equal to zero and solve for x:
x - 3 = 0 or 3x - 15 = 0
Solving the first equation, we add 3 to both sides:
x = 3
Solving the second equation, we add 15 to both sides and divide by 3:
3x = 15
x = 5
So, the solutions to the equation 3x^2 + 45 = 24x are x = 3 and x = 5.
To check our answers, we substitute these values back into the original equation and verify if both sides are equal:
For x = 3:
3(3)^2 + 45 = 24(3)
27 + 45 = 72
72 = 72 (True)
For x = 5:
3(5)^2 + 45 = 24(5)
75 + 45 = 120
120 = 120 (True)
Both solutions satisfy the original equation, so our answers are correct.