Solved on Jan 20, 2024

Solve the quadratic equation $0=-2(5 x-1)(3 x+4)$ for real values of $x$ .

STEP 1

Assumptions
1. We are given the equation $0=-2(5x-1)(3x+4)$ .
2. We need to solve for $x$ .

STEP 2

To solve for $x$ , we need to set each factor in the equation equal to zero and solve for $x$ in each case.
$-2(5x-1)(3x+4) = 0$

STEP 3

Since the product of several factors equals zero, at least one of the factors must be zero. We can ignore the factor $-2$ since it is a non-zero constant and focus on the factors containing $x$ .

STEP 4

Set the first factor containing $x$ equal to zero and solve for $x$ .
$(5x-1) = 0$

STEP 5

Add $1$ to both sides of the equation to isolate the term with $x$ .
$5x-1+1 = 0+1$

STEP 6

Simplify both sides of the equation.
$5x = 1$

STEP 7

Divide both sides of the equation by $5$ to solve for $x$ .
$\frac{5x}{5} = \frac{1}{5}$

STEP 8

Simplify the equation to find the first solution for $x$ .
$x = \frac{1}{5}$

STEP 9

Now, set the second factor containing $x$ equal to zero and solve for $x$ .
$(3x+4) = 0$

STEP 10

Subtract $4$ from both sides of the equation to isolate the term with $x$ .
$3x+4-4 = 0-4$

STEP 11

Simplify both sides of the equation.
$3x = -4$

STEP 12

Divide both sides of the equation by $3$ to solve for $x$ .
$\frac{3x}{3} = \frac{-4}{3}$

STEP 13

Simplify the equation to find the second solution for $x$ .
$x = \frac{-4}{3}$
The solutions to the equation $0=-2(5x-1)(3x+4)$ are $x = \frac{1}{5}$ and $x = \frac{-4}{3}$ .

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Solve the quadratic equation $0=-2(5 x-1)(3 x+4)$ for real values of $x$ .

STEP 1

Assumptions
1. We are given the equation $0=-2(5x-1)(3x+4)$ .
2. We need to solve for $x$ .

STEP 2

To solve for $x$ , we need to set each factor in the equation equal to zero and solve for $x$ in each case.
$-2(5x-1)(3x+4) = 0$

STEP 3

Since the product of several factors equals zero, at least one of the factors must be zero. We can ignore the factor $-2$ since it is a non-zero constant and focus on the factors containing $x$ .

STEP 4

Set the first factor containing $x$ equal to zero and solve for $x$ .
$(5x-1) = 0$

STEP 5

Add $1$ to both sides of the equation to isolate the term with $x$ .
$5x-1+1 = 0+1$

STEP 6

Simplify both sides of the equation.
$5x = 1$

STEP 7

Divide both sides of the equation by $5$ to solve for $x$ .
$\frac{5x}{5} = \frac{1}{5}$

STEP 8

Simplify the equation to find the first solution for $x$ .
$x = \frac{1}{5}$

STEP 9

Now, set the second factor containing $x$ equal to zero and solve for $x$ .
$(3x+4) = 0$

STEP 10

Subtract $4$ from both sides of the equation to isolate the term with $x$ .
$3x+4-4 = 0-4$

STEP 11

Simplify both sides of the equation.
$3x = -4$

STEP 12

Divide both sides of the equation by $3$ to solve for $x$ .
$\frac{3x}{3} = \frac{-4}{3}$

STEP 13

Simplify the equation to find the second solution for $x$ .
$x = \frac{-4}{3}$
The solutions to the equation $0=-2(5x-1)(3x+4)$ are $x = \frac{1}{5}$ and $x = \frac{-4}{3}$ .

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Solve the quadratic equation 0=−2(5x−1)(3x+4)0=-2(5 x-1)(3 x+4)0=−2(5x−1)(3x+4) for real values of xxx.

STEP 1

Assumptions1. We are given the equation 0=−2(5x−1)(3x+4)0=-2(5x-1)(3x+4)0=−2(5x−1)(3x+4).2. We need to solve for xxx.

STEP 2

To solve for xxx, we need to set each factor in the equation equal to zero and solve for xxx in each case.−2(5x−1)(3x+4)=0-2(5x-1)(3x+4) = 0−2(5x−1)(3x+4)=0

STEP 3

Since the product of several factors equals zero, at least one of the factors must be zero. We can ignore the factor −2-2−2 since it is a non-zero constant and focus on the factors containing xxx.

STEP 4

Set the first factor containing xxx equal to zero and solve for xxx.(5x−1)=0(5x-1) = 0(5x−1)=0

STEP 5

Add 111 to both sides of the equation to isolate the term with xxx.5x−1+1=0+15x-1+1 = 0+15x−1+1=0+1

STEP 6

Simplify both sides of the equation.5x=15x = 15x=1

STEP 7

Divide both sides of the equation by 555 to solve for xxx.5x5=15\frac{5x}{5} = \frac{1}{5}55x​=51​

STEP 8

Simplify the equation to find the first solution for xxx.x=15x = \frac{1}{5}x=51​

STEP 9

Now, set the second factor containing xxx equal to zero and solve for xxx.(3x+4)=0(3x+4) = 0(3x+4)=0

STEP 10

Subtract 444 from both sides of the equation to isolate the term with xxx.3x+4−4=0−43x+4-4 = 0-43x+4−4=0−4

STEP 11

Simplify both sides of the equation.3x=−43x = -43x=−4

STEP 12

Divide both sides of the equation by 333 to solve for xxx.3x3=−43\frac{3x}{3} = \frac{-4}{3}33x​=3−4​

STEP 13

Simplify the equation to find the second solution for xxx.x=−43x = \frac{-4}{3}x=3−4​The solutions to the equation 0=−2(5x−1)(3x+4)0=-2(5x-1)(3x+4)0=−2(5x−1)(3x+4) are x=15x = \frac{1}{5}x=51​ and x=−43x = \frac{-4}{3}x=3−4​.

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Solve the quadratic equation $0=-2(5 x-1)(3 x+4)$ for real values of $x$ .

Assumptions
1. We are given the equation $0=-2(5x-1)(3x+4)$ .
2. We need to solve for $x$ .

To solve for $x$ , we need to set each factor in the equation equal to zero and solve for $x$ in each case.
$-2(5x-1)(3x+4) = 0$

Since the product of several factors equals zero, at least one of the factors must be zero. We can ignore the factor $-2$ since it is a non-zero constant and focus on the factors containing $x$ .

Set the first factor containing $x$ equal to zero and solve for $x$ .
$(5x-1) = 0$

Add $1$ to both sides of the equation to isolate the term with $x$ .
$5x-1+1 = 0+1$

Simplify both sides of the equation.
$5x = 1$

Divide both sides of the equation by $5$ to solve for $x$ .
$\frac{5x}{5} = \frac{1}{5}$

Simplify the equation to find the first solution for $x$ .
$x = \frac{1}{5}$

Now, set the second factor containing $x$ equal to zero and solve for $x$ .
$(3x+4) = 0$

Subtract $4$ from both sides of the equation to isolate the term with $x$ .
$3x+4-4 = 0-4$

Simplify both sides of the equation.
$3x = -4$

Divide both sides of the equation by $3$ to solve for $x$ .
$\frac{3x}{3} = \frac{-4}{3}$

Simplify the equation to find the second solution for $x$ .
$x = \frac{-4}{3}$
The solutions to the equation $0=-2(5x-1)(3x+4)$ are $x = \frac{1}{5}$ and $x = \frac{-4}{3}$ .