Solved on Dec 12, 2023

Find the value of $x$ that satisfies the quadratic equation $3 x^{2} + 6 x = -3$ .

STEP 1

Assumptions
1. We are given a quadratic equation in the form $ax^2 + bx + c = 0$ .
2. The coefficients are $a = 3$ , $b = 6$ , and $c = -3$ .
3. We need to find the values of $x$ that satisfy the equation $3x^2 + 6x = -3$ .

STEP 2

First, we need to rewrite the equation in the standard form of a quadratic equation.
$3x^2 + 6x + 3 = 0$

STEP 3

Now, we can attempt to factor the quadratic if possible. We look for two numbers that multiply to $a \cdot c = 3 \cdot 3 = 9$ and add up to $b = 6$ .

STEP 4

The two numbers that satisfy these conditions are $3$ and $3$ , since $3 \cdot 3 = 9$ and $3 + 3 = 6$ .

STEP 5

We can now factor the quadratic equation using these numbers.
$3x^2 + 3x + 3x + 3 = 0$

STEP 6

Next, we group the terms to factor by grouping.
$(3x^2 + 3x) + (3x + 3) = 0$

STEP 7

Factor out the common factors from each group.
$3x(x + 1) + 3(x + 1) = 0$

STEP 8

Since both terms have a common factor of $(x + 1)$ , we can factor it out.
$(3x + 3)(x + 1) = 0$

STEP 9

Now we have the product of two factors equal to zero. According to the zero product property, if the product of two factors is zero, at least one of the factors must be zero.

STEP 10

Set each factor equal to zero and solve for $x$ .
$3x + 3 = 0 \quad \text{or} \quad x + 1 = 0$

STEP 11

Solve the first equation for $x$ .
$3x = -3$

STEP 12

Divide both sides by $3$ to isolate $x$ .
$x = -1$

STEP 13

Solve the second equation for $x$ .
$x = -1$

STEP 14

We find that both factors give us the same solution for $x$ .
The solution to the equation $3x^2 + 6x = -3$ is $x = -1$ .

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Find the value of $x$ that satisfies the quadratic equation $3 x^{2} + 6 x = -3$ .

STEP 1

Assumptions
1. We are given a quadratic equation in the form $ax^2 + bx + c = 0$ .
2. The coefficients are $a = 3$ , $b = 6$ , and $c = -3$ .
3. We need to find the values of $x$ that satisfy the equation $3x^2 + 6x = -3$ .

STEP 2

First, we need to rewrite the equation in the standard form of a quadratic equation.
$3x^2 + 6x + 3 = 0$

STEP 3

Now, we can attempt to factor the quadratic if possible. We look for two numbers that multiply to $a \cdot c = 3 \cdot 3 = 9$ and add up to $b = 6$ .

STEP 4

The two numbers that satisfy these conditions are $3$ and $3$ , since $3 \cdot 3 = 9$ and $3 + 3 = 6$ .

STEP 5

We can now factor the quadratic equation using these numbers.
$3x^2 + 3x + 3x + 3 = 0$

STEP 6

Next, we group the terms to factor by grouping.
$(3x^2 + 3x) + (3x + 3) = 0$

STEP 7

Factor out the common factors from each group.
$3x(x + 1) + 3(x + 1) = 0$

STEP 8

Since both terms have a common factor of $(x + 1)$ , we can factor it out.
$(3x + 3)(x + 1) = 0$

STEP 9

Now we have the product of two factors equal to zero. According to the zero product property, if the product of two factors is zero, at least one of the factors must be zero.

STEP 10

Set each factor equal to zero and solve for $x$ .
$3x + 3 = 0 \quad \text{or} \quad x + 1 = 0$

STEP 11

Solve the first equation for $x$ .
$3x = -3$

STEP 12

Divide both sides by $3$ to isolate $x$ .
$x = -1$

STEP 13

Solve the second equation for $x$ .
$x = -1$

STEP 14

We find that both factors give us the same solution for $x$ .
The solution to the equation $3x^2 + 6x = -3$ is $x = -1$ .

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Find the value of xxx that satisfies the quadratic equation 3x2+6x=−33 x^{2} + 6 x = -33x2+6x=−3.

STEP 1

Assumptions1. We are given a quadratic equation in the form ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0.2. The coefficients are a=3a = 3a=3, b=6b = 6b=6, and c=−3c = -3c=−3.3. We need to find the values of xxx that satisfy the equation 3x2+6x=−33x^2 + 6x = -33x2+6x=−3.

STEP 2

First, we need to rewrite the equation in the standard form of a quadratic equation.3x2+6x+3=03x^2 + 6x + 3 = 03x2+6x+3=0

STEP 3

Now, we can attempt to factor the quadratic if possible. We look for two numbers that multiply to a⋅c=3⋅3=9a \cdot c = 3 \cdot 3 = 9a⋅c=3⋅3=9 and add up to b=6b = 6b=6.

STEP 4

The two numbers that satisfy these conditions are 333 and 333, since 3⋅3=93 \cdot 3 = 93⋅3=9 and 3+3=63 + 3 = 63+3=6.

STEP 5

We can now factor the quadratic equation using these numbers.3x2+3x+3x+3=03x^2 + 3x + 3x + 3 = 03x2+3x+3x+3=0

STEP 6

Next, we group the terms to factor by grouping.(3x2+3x)+(3x+3)=0(3x^2 + 3x) + (3x + 3) = 0(3x2+3x)+(3x+3)=0

STEP 7

Factor out the common factors from each group.3x(x+1)+3(x+1)=03x(x + 1) + 3(x + 1) = 03x(x+1)+3(x+1)=0

STEP 8

Since both terms have a common factor of (x+1)(x + 1)(x+1), we can factor it out.(3x+3)(x+1)=0(3x + 3)(x + 1) = 0(3x+3)(x+1)=0

STEP 9

Now we have the product of two factors equal to zero. According to the zero product property, if the product of two factors is zero, at least one of the factors must be zero.

STEP 10

Set each factor equal to zero and solve for xxx.3x+3=0orx+1=03x + 3 = 0 \quad \text{or} \quad x + 1 = 03x+3=0orx+1=0

STEP 11

Solve the first equation for xxx.3x=−33x = -33x=−3

STEP 12

Divide both sides by 333 to isolate xxx.x=−1x = -1x=−1

STEP 13

Solve the second equation for xxx.x=−1x = -1x=−1

STEP 14

We find that both factors give us the same solution for xxx.The solution to the equation 3x2+6x=−33x^2 + 6x = -33x2+6x=−3 is x=−1x = -1x=−1.

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Find the value of $x$ that satisfies the quadratic equation $3 x^{2} + 6 x = -3$ .

Assumptions
1. We are given a quadratic equation in the form $ax^2 + bx + c = 0$ .
2. The coefficients are $a = 3$ , $b = 6$ , and $c = -3$ .
3. We need to find the values of $x$ that satisfy the equation $3x^2 + 6x = -3$ .

First, we need to rewrite the equation in the standard form of a quadratic equation.
$3x^2 + 6x + 3 = 0$

Now, we can attempt to factor the quadratic if possible. We look for two numbers that multiply to $a \cdot c = 3 \cdot 3 = 9$ and add up to $b = 6$ .

The two numbers that satisfy these conditions are $3$ and $3$ , since $3 \cdot 3 = 9$ and $3 + 3 = 6$ .

We can now factor the quadratic equation using these numbers.
$3x^2 + 3x + 3x + 3 = 0$

Next, we group the terms to factor by grouping.
$(3x^2 + 3x) + (3x + 3) = 0$

Factor out the common factors from each group.
$3x(x + 1) + 3(x + 1) = 0$

Since both terms have a common factor of $(x + 1)$ , we can factor it out.
$(3x + 3)(x + 1) = 0$

Set each factor equal to zero and solve for $x$ .
$3x + 3 = 0 \quad \text{or} \quad x + 1 = 0$

Solve the first equation for $x$ .
$3x = -3$

Divide both sides by $3$ to isolate $x$ .
$x = -1$

Solve the second equation for $x$ .
$x = -1$

We find that both factors give us the same solution for $x$ .
The solution to the equation $3x^2 + 6x = -3$ is $x = -1$ .