Solved on Jan 23, 2024

Solve the quadratic equation $7(5 x-4)^{2}+7=350$ for $x$ .

STEP 1

Assumptions
1. The given equation is $7(5x - 4)^2 + 7 = 350$ .
2. We need to solve for the variable $x$ .

STEP 2

First, we will isolate the term containing the variable $x$ by subtracting 7 from both sides of the equation.
$7(5x - 4)^2 + 7 - 7 = 350 - 7$

STEP 3

Simplify both sides of the equation.
$7(5x - 4)^2 = 343$

STEP 4

Now, divide both sides of the equation by 7 to further isolate the term with $x$ .
$\frac{7(5x - 4)^2}{7} = \frac{343}{7}$

STEP 5

Simplify the equation by performing the division.
$(5x - 4)^2 = 49$

STEP 6

To solve for $x$ , we need to take the square root of both sides of the equation. Remember that taking the square root of a number yields two solutions, one positive and one negative.
$\sqrt{(5x - 4)^2} = \pm\sqrt{49}$

STEP 7

Simplify the equation by taking the square root.
$5x - 4 = \pm7$

STEP 8

Now we have two separate equations to solve for $x$ :
1. $5x - 4 = 7$
2. $5x - 4 = -7$
We will solve each one separately.

STEP 9

First, let's solve $5x - 4 = 7$ .
Add 4 to both sides of the equation to isolate the term with $x$ .
$5x - 4 + 4 = 7 + 4$

STEP 10

Simplify the equation.
$5x = 11$

STEP 11

Divide both sides of the equation by 5 to solve for $x$ .
$\frac{5x}{5} = \frac{11}{5}$

STEP 12

Simplify the equation to find the first solution for $x$ .
$x = \frac{11}{5}$

STEP 13

Now, let's solve the second equation $5x - 4 = -7$ .
Add 4 to both sides of the equation to isolate the term with $x$ .
$5x - 4 + 4 = -7 + 4$

STEP 14

Simplify the equation.
$5x = -3$

STEP 15

Divide both sides of the equation by 5 to solve for $x$ .
$\frac{5x}{5} = \frac{-3}{5}$

STEP 16

Simplify the equation to find the second solution for $x$ .
$x = \frac{-3}{5}$
The solutions for the equation $7(5x - 4)^2 + 7 = 350$ are $x = \frac{11}{5}$ and $x = \frac{-3}{5}$ .

Was this helpful?

Solve the quadratic equation $7(5 x-4)^{2}+7=350$ for $x$ .

STEP 1

Assumptions
1. The given equation is $7(5x - 4)^2 + 7 = 350$ .
2. We need to solve for the variable $x$ .

STEP 2

First, we will isolate the term containing the variable $x$ by subtracting 7 from both sides of the equation.
$7(5x - 4)^2 + 7 - 7 = 350 - 7$

STEP 3

Simplify both sides of the equation.
$7(5x - 4)^2 = 343$

STEP 4

Now, divide both sides of the equation by 7 to further isolate the term with $x$ .
$\frac{7(5x - 4)^2}{7} = \frac{343}{7}$

STEP 5

Simplify the equation by performing the division.
$(5x - 4)^2 = 49$

STEP 6

To solve for $x$ , we need to take the square root of both sides of the equation. Remember that taking the square root of a number yields two solutions, one positive and one negative.
$\sqrt{(5x - 4)^2} = \pm\sqrt{49}$

STEP 7

Simplify the equation by taking the square root.
$5x - 4 = \pm7$

STEP 8

Now we have two separate equations to solve for $x$ :
1. $5x - 4 = 7$
2. $5x - 4 = -7$
We will solve each one separately.

STEP 9

First, let's solve $5x - 4 = 7$ .
Add 4 to both sides of the equation to isolate the term with $x$ .
$5x - 4 + 4 = 7 + 4$

STEP 10

Simplify the equation.
$5x = 11$

STEP 11

Divide both sides of the equation by 5 to solve for $x$ .
$\frac{5x}{5} = \frac{11}{5}$

STEP 12

Simplify the equation to find the first solution for $x$ .
$x = \frac{11}{5}$

STEP 13

Now, let's solve the second equation $5x - 4 = -7$ .
Add 4 to both sides of the equation to isolate the term with $x$ .
$5x - 4 + 4 = -7 + 4$

STEP 14

Simplify the equation.
$5x = -3$

STEP 15

Divide both sides of the equation by 5 to solve for $x$ .
$\frac{5x}{5} = \frac{-3}{5}$

STEP 16

Simplify the equation to find the second solution for $x$ .
$x = \frac{-3}{5}$
The solutions for the equation $7(5x - 4)^2 + 7 = 350$ are $x = \frac{11}{5}$ and $x = \frac{-3}{5}$ .

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Solve the quadratic equation 7(5x−4)2+7=3507(5 x-4)^{2}+7=3507(5x−4)2+7=350 for xxx.

STEP 1

Assumptions1. The given equation is 7(5x−4)2+7=3507(5x - 4)^2 + 7 = 3507(5x−4)2+7=350.2. We need to solve for the variable xxx.

STEP 2

First, we will isolate the term containing the variable xxx by subtracting 7 from both sides of the equation.7(5x−4)2+7−7=350−77(5x - 4)^2 + 7 - 7 = 350 - 77(5x−4)2+7−7=350−7

STEP 3

Simplify both sides of the equation.7(5x−4)2=3437(5x - 4)^2 = 3437(5x−4)2=343

STEP 4

Now, divide both sides of the equation by 7 to further isolate the term with xxx.7(5x−4)27=3437\frac{7(5x - 4)^2}{7} = \frac{343}{7}77(5x−4)2​=7343​

STEP 5

Simplify the equation by performing the division.(5x−4)2=49(5x - 4)^2 = 49(5x−4)2=49

STEP 6

To solve for xxx, we need to take the square root of both sides of the equation. Remember that taking the square root of a number yields two solutions, one positive and one negative.(5x−4)2=±49\sqrt{(5x - 4)^2} = \pm\sqrt{49}(5x−4)2​=±49​

STEP 7

Simplify the equation by taking the square root.5x−4=±75x - 4 = \pm75x−4=±7

STEP 8

Now we have two separate equations to solve for xxx:1. 5x−4=75x - 4 = 75x−4=72. 5x−4=−75x - 4 = -75x−4=−7We will solve each one separately.

STEP 9

First, let's solve 5x−4=75x - 4 = 75x−4=7.Add 4 to both sides of the equation to isolate the term with xxx.5x−4+4=7+45x - 4 + 4 = 7 + 45x−4+4=7+4

STEP 10

Simplify the equation.5x=115x = 115x=11

STEP 11

Divide both sides of the equation by 5 to solve for xxx.5x5=115\frac{5x}{5} = \frac{11}{5}55x​=511​

STEP 12

Simplify the equation to find the first solution for xxx.x=115x = \frac{11}{5}x=511​

STEP 13

Now, let's solve the second equation 5x−4=−75x - 4 = -75x−4=−7.Add 4 to both sides of the equation to isolate the term with xxx.5x−4+4=−7+45x - 4 + 4 = -7 + 45x−4+4=−7+4

STEP 14

Simplify the equation.5x=−35x = -35x=−3

STEP 15

Divide both sides of the equation by 5 to solve for xxx.5x5=−35\frac{5x}{5} = \frac{-3}{5}55x​=5−3​

STEP 16

Simplify the equation to find the second solution for xxx.x=−35x = \frac{-3}{5}x=5−3​The solutions for the equation 7(5x−4)2+7=3507(5x - 4)^2 + 7 = 3507(5x−4)2+7=350 are x=115x = \frac{11}{5}x=511​ and x=−35x = \frac{-3}{5}x=5−3​.

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Solve the quadratic equation $7(5 x-4)^{2}+7=350$ for $x$ .

Assumptions
1. The given equation is $7(5x - 4)^2 + 7 = 350$ .
2. We need to solve for the variable $x$ .

First, we will isolate the term containing the variable $x$ by subtracting 7 from both sides of the equation.
$7(5x - 4)^2 + 7 - 7 = 350 - 7$

Simplify both sides of the equation.
$7(5x - 4)^2 = 343$

Now, divide both sides of the equation by 7 to further isolate the term with $x$ .
$\frac{7(5x - 4)^2}{7} = \frac{343}{7}$

Simplify the equation by performing the division.
$(5x - 4)^2 = 49$

To solve for $x$ , we need to take the square root of both sides of the equation. Remember that taking the square root of a number yields two solutions, one positive and one negative.
$\sqrt{(5x - 4)^2} = \pm\sqrt{49}$

Simplify the equation by taking the square root.
$5x - 4 = \pm7$

Now we have two separate equations to solve for $x$ :
1. $5x - 4 = 7$
2. $5x - 4 = -7$
We will solve each one separately.

First, let's solve $5x - 4 = 7$ .
Add 4 to both sides of the equation to isolate the term with $x$ .
$5x - 4 + 4 = 7 + 4$

Simplify the equation.
$5x = 11$

Divide both sides of the equation by 5 to solve for $x$ .
$\frac{5x}{5} = \frac{11}{5}$

Simplify the equation to find the first solution for $x$ .
$x = \frac{11}{5}$

Now, let's solve the second equation $5x - 4 = -7$ .
Add 4 to both sides of the equation to isolate the term with $x$ .
$5x - 4 + 4 = -7 + 4$

Simplify the equation.
$5x = -3$

Divide both sides of the equation by 5 to solve for $x$ .
$\frac{5x}{5} = \frac{-3}{5}$

Simplify the equation to find the second solution for $x$ .
$x = \frac{-3}{5}$
The solutions for the equation $7(5x - 4)^2 + 7 = 350$ are $x = \frac{11}{5}$ and $x = \frac{-3}{5}$ .