Solved on Dec 10, 2023

Solve the quadratic equation $6n^2 + 42n - 48 = 6$ for the value of $n$ .

STEP 1

Assumptions
1. We are given a quadratic equation in the form $6n^2 + 42n - 48 = 6$ .
2. We need to solve for the variable $n$ .

STEP 2

First, we need to simplify the equation by moving all terms to one side to set the equation to zero.
$6n^2 + 42n - 48 - 6 = 0$

STEP 3

Subtract 6 from both sides of the equation to combine like terms.
$6n^2 + 42n - 54 = 0$

STEP 4

Now, we can try to factor the quadratic equation. To do this, we look for two numbers that multiply to $6 \times -54$ (the coefficient of $n^2$ times the constant term) and add to $42$ (the coefficient of $n$ ).

STEP 5

The two numbers that satisfy these conditions are $54$ and $-6$ because $54 \times -6 = -324$ (which is $6 \times -54$ ) and $54 + (-6) = 48$ (which is close to $42$ , but we need to consider the factor of $6$ in the equation).

STEP 6

We rewrite the middle term $42n$ using the numbers $54$ and $-6$ .
$6n^2 + 54n - 6n - 54 = 0$

STEP 7

Now, we can factor by grouping. Group the first two terms together and the last two terms together.
$(6n^2 + 54n) + (-6n - 54) = 0$

STEP 8

Factor out the greatest common factor from each group.
$6n(n + 9) - 6(n + 9) = 0$

STEP 9

Since both groups contain the factor $(n + 9)$ , we can factor it out.
$(6n - 6)(n + 9) = 0$

STEP 10

We can further factor out a $6$ from the first term.
$6(n - 1)(n + 9) = 0$

STEP 11

Now, apply the zero-product property, which states that if a product of factors equals zero, then at least one of the factors must be zero.
$6 = 0, \quad n - 1 = 0, \quad \text{or} \quad n + 9 = 0$

STEP 12

Since $6$ cannot equal $0$ , we can discard that part and solve the remaining two equations for $n$ .
$n - 1 = 0$
$n + 9 = 0$

STEP 13

Solve the first equation for $n$ .
$n = 1$

STEP 14

Solve the second equation for $n$ .
$n = -9$
The solutions to the quadratic equation are $n = 1$ and $n = -9$ .

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Solve the quadratic equation $6n^2 + 42n - 48 = 6$ for the value of $n$ .

STEP 1

Assumptions
1. We are given a quadratic equation in the form $6n^2 + 42n - 48 = 6$ .
2. We need to solve for the variable $n$ .

STEP 2

First, we need to simplify the equation by moving all terms to one side to set the equation to zero.
$6n^2 + 42n - 48 - 6 = 0$

STEP 3

Subtract 6 from both sides of the equation to combine like terms.
$6n^2 + 42n - 54 = 0$

STEP 4

Now, we can try to factor the quadratic equation. To do this, we look for two numbers that multiply to $6 \times -54$ (the coefficient of $n^2$ times the constant term) and add to $42$ (the coefficient of $n$ ).

STEP 5

The two numbers that satisfy these conditions are $54$ and $-6$ because $54 \times -6 = -324$ (which is $6 \times -54$ ) and $54 + (-6) = 48$ (which is close to $42$ , but we need to consider the factor of $6$ in the equation).

STEP 6

We rewrite the middle term $42n$ using the numbers $54$ and $-6$ .
$6n^2 + 54n - 6n - 54 = 0$

STEP 7

Now, we can factor by grouping. Group the first two terms together and the last two terms together.
$(6n^2 + 54n) + (-6n - 54) = 0$

STEP 8

Factor out the greatest common factor from each group.
$6n(n + 9) - 6(n + 9) = 0$

STEP 9

Since both groups contain the factor $(n + 9)$ , we can factor it out.
$(6n - 6)(n + 9) = 0$

STEP 10

We can further factor out a $6$ from the first term.
$6(n - 1)(n + 9) = 0$

STEP 11

Now, apply the zero-product property, which states that if a product of factors equals zero, then at least one of the factors must be zero.
$6 = 0, \quad n - 1 = 0, \quad \text{or} \quad n + 9 = 0$

STEP 12

Since $6$ cannot equal $0$ , we can discard that part and solve the remaining two equations for $n$ .
$n - 1 = 0$
$n + 9 = 0$

STEP 13

Solve the first equation for $n$ .
$n = 1$

STEP 14

Solve the second equation for $n$ .
$n = -9$
The solutions to the quadratic equation are $n = 1$ and $n = -9$ .

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Solve the quadratic equation 6n2+42n−48=66n^2 + 42n - 48 = 66n2+42n−48=6 for the value of nnn.

STEP 1

Assumptions1. We are given a quadratic equation in the form 6n2+42n−48=66n^2 + 42n - 48 = 66n2+42n−48=6.2. We need to solve for the variable nnn.

STEP 2

First, we need to simplify the equation by moving all terms to one side to set the equation to zero.6n2+42n−48−6=06n^2 + 42n - 48 - 6 = 06n2+42n−48−6=0

STEP 3

Subtract 6 from both sides of the equation to combine like terms.6n2+42n−54=06n^2 + 42n - 54 = 06n2+42n−54=0

STEP 4

Now, we can try to factor the quadratic equation. To do this, we look for two numbers that multiply to 6×−546 \times -546×−54 (the coefficient of n2n^2n2 times the constant term) and add to 424242 (the coefficient of nnn).

STEP 5

The two numbers that satisfy these conditions are 545454 and −6-6−6 because 54×−6=−32454 \times -6 = -32454×−6=−324 (which is 6×−546 \times -546×−54) and 54+(−6)=4854 + (-6) = 4854+(−6)=48 (which is close to 424242, but we need to consider the factor of 666 in the equation).

STEP 6

We rewrite the middle term 42n42n42n using the numbers 545454 and −6-6−6.6n2+54n−6n−54=06n^2 + 54n - 6n - 54 = 06n2+54n−6n−54=0

STEP 7

Now, we can factor by grouping. Group the first two terms together and the last two terms together.(6n2+54n)+(−6n−54)=0(6n^2 + 54n) + (-6n - 54) = 0(6n2+54n)+(−6n−54)=0

STEP 8

Factor out the greatest common factor from each group.6n(n+9)−6(n+9)=06n(n + 9) - 6(n + 9) = 06n(n+9)−6(n+9)=0

STEP 9

Since both groups contain the factor (n+9)(n + 9)(n+9), we can factor it out.(6n−6)(n+9)=0(6n - 6)(n + 9) = 0(6n−6)(n+9)=0

STEP 10

We can further factor out a 666 from the first term.6(n−1)(n+9)=06(n - 1)(n + 9) = 06(n−1)(n+9)=0

STEP 11

Now, apply the zero-product property, which states that if a product of factors equals zero, then at least one of the factors must be zero.6=0,n−1=0,orn+9=06 = 0, \quad n - 1 = 0, \quad \text{or} \quad n + 9 = 06=0,n−1=0,orn+9=0

STEP 12

Since 666 cannot equal 000, we can discard that part and solve the remaining two equations for nnn.n−1=0n - 1 = 0n−1=0n+9=0n + 9 = 0n+9=0

STEP 13

Solve the first equation for nnn.n=1n = 1n=1

STEP 14

Solve the second equation for nnn.n=−9n = -9n=−9The solutions to the quadratic equation are n=1n = 1n=1 and n=−9n = -9n=−9.

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Solve the quadratic equation $6n^2 + 42n - 48 = 6$ for the value of $n$ .

Assumptions
1. We are given a quadratic equation in the form $6n^2 + 42n - 48 = 6$ .
2. We need to solve for the variable $n$ .

First, we need to simplify the equation by moving all terms to one side to set the equation to zero.
$6n^2 + 42n - 48 - 6 = 0$

Subtract 6 from both sides of the equation to combine like terms.
$6n^2 + 42n - 54 = 0$

Now, we can try to factor the quadratic equation. To do this, we look for two numbers that multiply to $6 \times -54$ (the coefficient of $n^2$ times the constant term) and add to $42$ (the coefficient of $n$ ).

The two numbers that satisfy these conditions are $54$ and $-6$ because $54 \times -6 = -324$ (which is $6 \times -54$ ) and $54 + (-6) = 48$ (which is close to $42$ , but we need to consider the factor of $6$ in the equation).

We rewrite the middle term $42n$ using the numbers $54$ and $-6$ .
$6n^2 + 54n - 6n - 54 = 0$

Now, we can factor by grouping. Group the first two terms together and the last two terms together.
$(6n^2 + 54n) + (-6n - 54) = 0$

Factor out the greatest common factor from each group.
$6n(n + 9) - 6(n + 9) = 0$

Since both groups contain the factor $(n + 9)$ , we can factor it out.
$(6n - 6)(n + 9) = 0$

We can further factor out a $6$ from the first term.
$6(n - 1)(n + 9) = 0$

Now, apply the zero-product property, which states that if a product of factors equals zero, then at least one of the factors must be zero.
$6 = 0, \quad n - 1 = 0, \quad \text{or} \quad n + 9 = 0$

Since $6$ cannot equal $0$ , we can discard that part and solve the remaining two equations for $n$ .
$n - 1 = 0$
$n + 9 = 0$

Solve the first equation for $n$ .
$n = 1$

Solve the second equation for $n$ .
$n = -9$
The solutions to the quadratic equation are $n = 1$ and $n = -9$ .