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Answer :
Sure! Let's solve the equation step-by-step:
The equation we have is:
[tex]\[ 10x^3 + 45x^2 + 35x = 0 \][/tex]
### Step 1: Factor Out Common Terms
Notice that each term in the equation contains an [tex]\(x\)[/tex] and they are all divisible by 5. So, we can factor out the greatest common factor, which is [tex]\(5x\)[/tex]:
[tex]\[ 5x(2x^2 + 9x + 7) = 0 \][/tex]
### Step 2: Solve for [tex]\(x\)[/tex] when [tex]\(5x = 0\)[/tex]
When [tex]\(5x = 0\)[/tex], it's straightforward to solve for [tex]\(x\)[/tex]:
[tex]\[ x = 0 \][/tex]
This is one solution of the equation.
### Step 3: Solve the Quadratic Equation
Next, we need to solve the quadratic equation:
[tex]\[ 2x^2 + 9x + 7 = 0 \][/tex]
To solve this, we'll use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Where [tex]\(a = 2\)[/tex], [tex]\(b = 9\)[/tex], and [tex]\(c = 7\)[/tex].
### Step 4: Calculate the Discriminant
First, we calculate the discriminant:
[tex]\[ b^2 - 4ac = 9^2 - 4 \times 2 \times 7 = 81 - 56 = 25 \][/tex]
Since the discriminant is positive, there are two real roots.
### Step 5: Calculate the Roots
Now, substitute the values into the quadratic formula:
1. First root:
[tex]\[ x = \frac{-9 + \sqrt{25}}{4} = \frac{-9 + 5}{4} = \frac{-4}{4} = -1.0 \][/tex]
2. Second root:
[tex]\[ x = \frac{-9 - \sqrt{25}}{4} = \frac{-9 - 5}{4} = \frac{-14}{4} = -3.5 \][/tex]
### Final Solutions
Thus, the solutions to the equation [tex]\(10x^3 + 45x^2 + 35x = 0\)[/tex] are:
1. [tex]\(x = 0\)[/tex]
2. [tex]\(x = -1.0\)[/tex]
3. [tex]\(x = -3.5\)[/tex]
These are the values of [tex]\(x\)[/tex] that satisfy the given equation.
The equation we have is:
[tex]\[ 10x^3 + 45x^2 + 35x = 0 \][/tex]
### Step 1: Factor Out Common Terms
Notice that each term in the equation contains an [tex]\(x\)[/tex] and they are all divisible by 5. So, we can factor out the greatest common factor, which is [tex]\(5x\)[/tex]:
[tex]\[ 5x(2x^2 + 9x + 7) = 0 \][/tex]
### Step 2: Solve for [tex]\(x\)[/tex] when [tex]\(5x = 0\)[/tex]
When [tex]\(5x = 0\)[/tex], it's straightforward to solve for [tex]\(x\)[/tex]:
[tex]\[ x = 0 \][/tex]
This is one solution of the equation.
### Step 3: Solve the Quadratic Equation
Next, we need to solve the quadratic equation:
[tex]\[ 2x^2 + 9x + 7 = 0 \][/tex]
To solve this, we'll use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Where [tex]\(a = 2\)[/tex], [tex]\(b = 9\)[/tex], and [tex]\(c = 7\)[/tex].
### Step 4: Calculate the Discriminant
First, we calculate the discriminant:
[tex]\[ b^2 - 4ac = 9^2 - 4 \times 2 \times 7 = 81 - 56 = 25 \][/tex]
Since the discriminant is positive, there are two real roots.
### Step 5: Calculate the Roots
Now, substitute the values into the quadratic formula:
1. First root:
[tex]\[ x = \frac{-9 + \sqrt{25}}{4} = \frac{-9 + 5}{4} = \frac{-4}{4} = -1.0 \][/tex]
2. Second root:
[tex]\[ x = \frac{-9 - \sqrt{25}}{4} = \frac{-9 - 5}{4} = \frac{-14}{4} = -3.5 \][/tex]
### Final Solutions
Thus, the solutions to the equation [tex]\(10x^3 + 45x^2 + 35x = 0\)[/tex] are:
1. [tex]\(x = 0\)[/tex]
2. [tex]\(x = -1.0\)[/tex]
3. [tex]\(x = -3.5\)[/tex]
These are the values of [tex]\(x\)[/tex] that satisfy the given equation.
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