Question
Solve the equation for all values of xx.
minus, x, left parenthesis, x, minus, 9, right parenthesis, left parenthesis, 4, x, plus, 9, right parenthesis, equals, 0
−x(x−9)(4x+9)=0
minus, x, left parenthesis, x, minus, 9, right parenthesis, left parenthesis, 4, x, plus, 9, right parenthesis, equals, 0
−x(x−9)(4x+9)=0
Answers
Answer
Write two numbers that multiply to the value on top and add to the value on bottom.
3636
1212
times×
plus+
3636
1212
times×
plus+
Answer
sorr the numbers are actully 36 and 12
Answer
Write two numbers that multiply to the value on top and add to the value on bottom.
36 - on top
12 - on bottem
times× - on top
plus+ - on bottem
36 - on top
12 - on bottem
times× - on top
plus+ - on bottem
Answer
Factor.
x, squared, minus, 2, x, minus, 8
x
2
−2x−8
x, squared, minus, 2, x, minus, 8
x
2
−2x−8
Answer
solve for all values of xx by factoring.
x, squared, plus, 8, x, minus, 24, equals, 3, x
x
2
+8x−24=
3x
x, squared, plus, 8, x, minus, 24, equals, 3, x
x
2
+8x−24=
3x
Answered by
GPT-4o mini
To solve the equation
\[
-x(x - 9)(4x + 9) = 0,
\]
we can set each factor equal to zero.
1. The first factor is \(-x\):
\[
-x = 0 \quad \Rightarrow \quad x = 0.
\]
2. The second factor is \(x - 9\):
\[
x - 9 = 0 \quad \Rightarrow \quad x = 9.
\]
3. The third factor is \(4x + 9\):
\[
4x + 9 = 0 \quad \Rightarrow \quad 4x = -9 \quad \Rightarrow \quad x = -\frac{9}{4}.
\]
Now, we can list all the solutions we found:
\[
x = 0, \quad x = 9, \quad x = -\frac{9}{4}.
\]
Therefore, the complete set of solutions to the equation is:
\[
\boxed{0, 9, -\frac{9}{4}}.
\]
\[
-x(x - 9)(4x + 9) = 0,
\]
we can set each factor equal to zero.
1. The first factor is \(-x\):
\[
-x = 0 \quad \Rightarrow \quad x = 0.
\]
2. The second factor is \(x - 9\):
\[
x - 9 = 0 \quad \Rightarrow \quad x = 9.
\]
3. The third factor is \(4x + 9\):
\[
4x + 9 = 0 \quad \Rightarrow \quad 4x = -9 \quad \Rightarrow \quad x = -\frac{9}{4}.
\]
Now, we can list all the solutions we found:
\[
x = 0, \quad x = 9, \quad x = -\frac{9}{4}.
\]
Therefore, the complete set of solutions to the equation is:
\[
\boxed{0, 9, -\frac{9}{4}}.
\]
Answered by
GPT-4o mini
To find two numbers that multiply to \( 3636 \) and add up to \( 1212 \), we can set up the following equations:
Let the two numbers be \( a \) and \( b \).
1. \( a \times b = 3636 \)
2. \( a + b = 1212 \)
To solve this, we can express \( b \) in terms of \( a \) using the second equation:
\[
b = 1212 - a
\]
Now, substitute this expression for \( b \) into the first equation:
\[
a \times (1212 - a) = 3636
\]
Expanding this gives:
\[
1212a - a^2 = 3636
\]
Rearranging the equation:
\[
a^2 - 1212a + 3636 = 0
\]
We can use the quadratic formula to solve for \( a \):
\[
a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1 \), \( b = -1212 \), and \( c = 3636 \).
Calculating the discriminant:
\[
b^2 - 4ac = (-1212)^2 - 4 \cdot 1 \cdot 3636
\]
\[
= 1460544 - 1454400
\]
\[
= 6144
\]
Now substituting back into the quadratic formula:
\[
a = \frac{1212 \pm \sqrt{6144}}{2}
\]
Calculating \( \sqrt{6144} \):
\[
\sqrt{6144} \approx 78.4
\]
Now, substituting this back in:
\[
a = \frac{1212 \pm 78.4}{2}
\]
Calculating both possible values:
1. Using the plus sign:
\[
a = \frac{1212 + 78.4}{2} \approx \frac{1290.4}{2} \approx 645.2
\]
2. Using the minus sign:
\[
a = \frac{1212 - 78.4}{2} \approx \frac{1133.6}{2} \approx 566.8
\]
Now let's find \( b \):
If \( a \approx 645.2 \), then
\[
b \approx 1212 - 645.2 \approx 566.8
\]
Therefore, the two numbers that multiply to \( 3636 \) and add up to \( 1212 \) are approximately:
\[
\boxed{645.2 \text{ and } 566.8}
\]
However, it seems that since these are not integers, it is best to find precise integers. Let's find the two numbers through factor pairs of \( 3636 \) since they must be whole numbers.
The pairs of factors of \( 3636 \) include \( (1, 3636), (2, 1818), (3, 1212), (6, 606), (9, 404), (18, 202), (36, 101), (2, 1818) \), and we look for ones that add to \( 1212 \):
Testing pairs of factors:
- \( 606 + 606 = 1212 \) and \( 606 \times 6 = 3636\).
So the exact numbers you're looking for are:
\[
\boxed{606 \text{ and } 606}
\]
Let the two numbers be \( a \) and \( b \).
1. \( a \times b = 3636 \)
2. \( a + b = 1212 \)
To solve this, we can express \( b \) in terms of \( a \) using the second equation:
\[
b = 1212 - a
\]
Now, substitute this expression for \( b \) into the first equation:
\[
a \times (1212 - a) = 3636
\]
Expanding this gives:
\[
1212a - a^2 = 3636
\]
Rearranging the equation:
\[
a^2 - 1212a + 3636 = 0
\]
We can use the quadratic formula to solve for \( a \):
\[
a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1 \), \( b = -1212 \), and \( c = 3636 \).
Calculating the discriminant:
\[
b^2 - 4ac = (-1212)^2 - 4 \cdot 1 \cdot 3636
\]
\[
= 1460544 - 1454400
\]
\[
= 6144
\]
Now substituting back into the quadratic formula:
\[
a = \frac{1212 \pm \sqrt{6144}}{2}
\]
Calculating \( \sqrt{6144} \):
\[
\sqrt{6144} \approx 78.4
\]
Now, substituting this back in:
\[
a = \frac{1212 \pm 78.4}{2}
\]
Calculating both possible values:
1. Using the plus sign:
\[
a = \frac{1212 + 78.4}{2} \approx \frac{1290.4}{2} \approx 645.2
\]
2. Using the minus sign:
\[
a = \frac{1212 - 78.4}{2} \approx \frac{1133.6}{2} \approx 566.8
\]
Now let's find \( b \):
If \( a \approx 645.2 \), then
\[
b \approx 1212 - 645.2 \approx 566.8
\]
Therefore, the two numbers that multiply to \( 3636 \) and add up to \( 1212 \) are approximately:
\[
\boxed{645.2 \text{ and } 566.8}
\]
However, it seems that since these are not integers, it is best to find precise integers. Let's find the two numbers through factor pairs of \( 3636 \) since they must be whole numbers.
The pairs of factors of \( 3636 \) include \( (1, 3636), (2, 1818), (3, 1212), (6, 606), (9, 404), (18, 202), (36, 101), (2, 1818) \), and we look for ones that add to \( 1212 \):
Testing pairs of factors:
- \( 606 + 606 = 1212 \) and \( 606 \times 6 = 3636\).
So the exact numbers you're looking for are:
\[
\boxed{606 \text{ and } 606}
\]
Answered by
GPT-4o mini
Let's verify the numbers \(36\) and \(12\) based on your criteria.
1. Multiply:
\[
36 \times 12 = 432
\]
2. Add:
\[
36 + 12 = 48
\]
It seems that \(36\) and \(12\) do not multiply to \(3636\) or add to \(1212\).
If you meant to find two specific numbers that multiply to \(432\) and add to \(48\), then \(36\) and \(12\) do indeed fit those requirements.
If you would like to give the original numbers again, or need help with a different problem, please let me know!
1. Multiply:
\[
36 \times 12 = 432
\]
2. Add:
\[
36 + 12 = 48
\]
It seems that \(36\) and \(12\) do not multiply to \(3636\) or add to \(1212\).
If you meant to find two specific numbers that multiply to \(432\) and add to \(48\), then \(36\) and \(12\) do indeed fit those requirements.
If you would like to give the original numbers again, or need help with a different problem, please let me know!
Answered by
GPT-4o mini
To find two numbers that multiply to \(36\) (the value on top) and add to \(12\) (the value on the bottom), we can denote the two numbers as \(x\) and \(y\).
1. The first equation (for multiplication):
\[
x \times y = 36
\]
2. The second equation (for addition):
\[
x + y = 12
\]
We can express \(y\) in terms of \(x\) using the second equation:
\[
y = 12 - x
\]
Now, we can substitute this expression for \(y\) into the first equation:
\[
x(12 - x) = 36
\]
Expanding this gives:
\[
12x - x^2 = 36
\]
Rearranging the equation:
\[
x^2 - 12x + 36 = 0
\]
This quadratic equation can be factored easily:
\[
(x - 6)(x - 6) = 0
\]
So, the solution is:
\[
x - 6 = 0 \quad \Rightarrow \quad x = 6
\]
Since both factors are the same:
\[
y = 6
\]
Thus, the two numbers that multiply to \(36\) and add to \(12\) are both:
\[
\boxed{6 \text{ and } 6}.
\]
1. The first equation (for multiplication):
\[
x \times y = 36
\]
2. The second equation (for addition):
\[
x + y = 12
\]
We can express \(y\) in terms of \(x\) using the second equation:
\[
y = 12 - x
\]
Now, we can substitute this expression for \(y\) into the first equation:
\[
x(12 - x) = 36
\]
Expanding this gives:
\[
12x - x^2 = 36
\]
Rearranging the equation:
\[
x^2 - 12x + 36 = 0
\]
This quadratic equation can be factored easily:
\[
(x - 6)(x - 6) = 0
\]
So, the solution is:
\[
x - 6 = 0 \quad \Rightarrow \quad x = 6
\]
Since both factors are the same:
\[
y = 6
\]
Thus, the two numbers that multiply to \(36\) and add to \(12\) are both:
\[
\boxed{6 \text{ and } 6}.
\]
Answered by
GPT-4o mini
To factor the quadratic expression
\[
x^2 - 2x - 8,
\]
we need to find two numbers that multiply to \(-8\) (the constant term) and add up to \(-2\) (the coefficient of the linear term).
The pairs of factors of \(-8\) are:
- \(1\) and \(-8\) → \(1 + (-8) = -7\)
- \(-1\) and \(8\) → \(-1 + 8 = 7\)
- \(2\) and \(-4\) → \(2 + (-4) = -2\)
- \(-2\) and \(4\) → \(-2 + 4 = 2\)
The pair that adds up to \(-2\) is \(2\) and \(-4\).
We can rewrite the quadratic as:
\[
x^2 - 4x + 2x - 8.
\]
Next, we group the terms:
\[
(x^2 - 4x) + (2x - 8).
\]
Factoring out the common factors in each group:
\[
x(x - 4) + 2(x - 4).
\]
Now we can factor out the common binomial factor \((x - 4)\):
\[
(x - 4)(x + 2).
\]
Thus, the factored form of the expression \(x^2 - 2x - 8\) is:
\[
\boxed{(x - 4)(x + 2)}.
\]
\[
x^2 - 2x - 8,
\]
we need to find two numbers that multiply to \(-8\) (the constant term) and add up to \(-2\) (the coefficient of the linear term).
The pairs of factors of \(-8\) are:
- \(1\) and \(-8\) → \(1 + (-8) = -7\)
- \(-1\) and \(8\) → \(-1 + 8 = 7\)
- \(2\) and \(-4\) → \(2 + (-4) = -2\)
- \(-2\) and \(4\) → \(-2 + 4 = 2\)
The pair that adds up to \(-2\) is \(2\) and \(-4\).
We can rewrite the quadratic as:
\[
x^2 - 4x + 2x - 8.
\]
Next, we group the terms:
\[
(x^2 - 4x) + (2x - 8).
\]
Factoring out the common factors in each group:
\[
x(x - 4) + 2(x - 4).
\]
Now we can factor out the common binomial factor \((x - 4)\):
\[
(x - 4)(x + 2).
\]
Thus, the factored form of the expression \(x^2 - 2x - 8\) is:
\[
\boxed{(x - 4)(x + 2)}.
\]
Answered by
GPT-4o mini
To solve the equation
\[
x^2 + 8x - 24 = 3x,
\]
we first set it to zero by moving all terms to one side of the equation:
\[
x^2 + 8x - 3x - 24 = 0.
\]
This simplifies to:
\[
x^2 + 5x - 24 = 0.
\]
Next, we factor the quadratic equation. We need to find two numbers that multiply to \(-24\) (the constant term) and add up to \(5\) (the coefficient of the linear term).
The pairs of factors of \(-24\) are:
- \(1\) and \(-24\) → \(1 - 24 = -23\)
- \(-1\) and \(24\) → \(-1 + 24 = 23\)
- \(2\) and \(-12\) → \(2 - 12 = -10\)
- \(-2\) and \(12\) → \(-2 + 12 = 10\)
- \(3\) and \(-8\) → \(3 - 8 = -5\)
- \(-3\) and \(8\) → \(-3 + 8 = 5\)
The pair that adds up to \(5\) is \(-3\) and \(8\).
So, we can rewrite the quadratic as:
\[
x^2 - 3x + 8x - 24 = 0.
\]
Next, we group the terms:
\[
(x^2 - 3x) + (8x - 24) = 0.
\]
Factoring out the common factors in each group gives:
\[
x(x - 3) + 8(x - 3) = 0.
\]
Now we can factor out the common binomial factor \((x - 3)\):
\[
(x - 3)(x + 8) = 0.
\]
Setting each factor to zero gives:
1. \(x - 3 = 0\)
\[
x = 3.
\]
2. \(x + 8 = 0\)
\[
x = -8.
\]
Thus, the solutions to the equation are:
\[
\boxed{3 \text{ and } -8}.
\]
\[
x^2 + 8x - 24 = 3x,
\]
we first set it to zero by moving all terms to one side of the equation:
\[
x^2 + 8x - 3x - 24 = 0.
\]
This simplifies to:
\[
x^2 + 5x - 24 = 0.
\]
Next, we factor the quadratic equation. We need to find two numbers that multiply to \(-24\) (the constant term) and add up to \(5\) (the coefficient of the linear term).
The pairs of factors of \(-24\) are:
- \(1\) and \(-24\) → \(1 - 24 = -23\)
- \(-1\) and \(24\) → \(-1 + 24 = 23\)
- \(2\) and \(-12\) → \(2 - 12 = -10\)
- \(-2\) and \(12\) → \(-2 + 12 = 10\)
- \(3\) and \(-8\) → \(3 - 8 = -5\)
- \(-3\) and \(8\) → \(-3 + 8 = 5\)
The pair that adds up to \(5\) is \(-3\) and \(8\).
So, we can rewrite the quadratic as:
\[
x^2 - 3x + 8x - 24 = 0.
\]
Next, we group the terms:
\[
(x^2 - 3x) + (8x - 24) = 0.
\]
Factoring out the common factors in each group gives:
\[
x(x - 3) + 8(x - 3) = 0.
\]
Now we can factor out the common binomial factor \((x - 3)\):
\[
(x - 3)(x + 8) = 0.
\]
Setting each factor to zero gives:
1. \(x - 3 = 0\)
\[
x = 3.
\]
2. \(x + 8 = 0\)
\[
x = -8.
\]
Thus, the solutions to the equation are:
\[
\boxed{3 \text{ and } -8}.
\]
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