Math / Algebra

QuestionSolve the following equation by factoring. $9 x^{2}+25=30 x$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is $\square$ \}. (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. There is no solution.

Studdy Solution

STEP 1

1. The equation $9x^2 + 25 = 30x$ is a quadratic equation.
2. Solving the equation involves rearranging it into standard quadratic form.
3. The quadratic can be solved by factoring.

STEP 2

1. Rearrange the equation into standard quadratic form.
2. Factor the quadratic expression.
3. Solve for the roots of the equation.
4. Determine the solution set.

STEP 3

Rearrange the equation to standard quadratic form $ax^2 + bx + c = 0$ . Subtract $30x$ from both sides:
$9x^2 + 25 = 30x$ $9x^2 - 30x + 25 = 0$

STEP 4

Factor the quadratic expression. Look for two numbers that multiply to $9 \times 25 = 225$ and add to $-30$ . The numbers are $-15$ and $-15$ .
$9x^2 - 30x + 25 = (3x - 5)(3x - 5)$

STEP 5

Solve for the roots by setting each factor equal to zero. Since both factors are the same, we only need to solve one equation:
$3x - 5 = 0$

STEP 6

Solve for $x$ :
$3x - 5 = 0$ $3x = 5$ $x = \frac{5}{3}$

STEP 7

Determine the solution set. Since both factors were the same, there is only one unique solution:
The solution set is $\left\{\frac{5}{3}\right\}$ .
The correct choice is: A. The solution set is $\left\{\frac{5}{3}\right\}$ .

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Studdy Solution

STEP 1

1. The equation $9x^2 + 25 = 30x$ is a quadratic equation.
2. Solving the equation involves rearranging it into standard quadratic form.
3. The quadratic can be solved by factoring.

STEP 2

1. Rearrange the equation into standard quadratic form.
2. Factor the quadratic expression.
3. Solve for the roots of the equation.
4. Determine the solution set.

STEP 3

Rearrange the equation to standard quadratic form $ax^2 + bx + c = 0$ . Subtract $30x$ from both sides:
$9x^2 + 25 = 30x$ $9x^2 - 30x + 25 = 0$

STEP 4

Factor the quadratic expression. Look for two numbers that multiply to $9 \times 25 = 225$ and add to $-30$ . The numbers are $-15$ and $-15$ .
$9x^2 - 30x + 25 = (3x - 5)(3x - 5)$

STEP 5

Solve for the roots by setting each factor equal to zero. Since both factors are the same, we only need to solve one equation:
$3x - 5 = 0$

STEP 6

Solve for $x$ :
$3x - 5 = 0$ $3x = 5$ $x = \frac{5}{3}$

STEP 7

Determine the solution set. Since both factors were the same, there is only one unique solution:
The solution set is $\left\{\frac{5}{3}\right\}$ .
The correct choice is: A. The solution set is $\left\{\frac{5}{3}\right\}$ .

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Studdy Solution

STEP 1

1. The equation 9x2+25=30x9x^2 + 25 = 30x9x2+25=30x is a quadratic equation.2. Solving the equation involves rearranging it into standard quadratic form.3. The quadratic can be solved by factoring.

STEP 2

1. Rearrange the equation into standard quadratic form.2. Factor the quadratic expression.3. Solve for the roots of the equation.4. Determine the solution set.

STEP 3

Rearrange the equation to standard quadratic form ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0. Subtract 30x30x30x from both sides:9x2+25=30x 9x^2 + 25 = 30x 9x2+25=30x 9x2−30x+25=0 9x^2 - 30x + 25 = 0 9x2−30x+25=0

STEP 4

Factor the quadratic expression. Look for two numbers that multiply to 9×25=2259 \times 25 = 2259×25=225 and add to −30-30−30. The numbers are −15-15−15 and −15-15−15.9x2−30x+25=(3x−5)(3x−5) 9x^2 - 30x + 25 = (3x - 5)(3x - 5) 9x2−30x+25=(3x−5)(3x−5)

STEP 5

Solve for the roots by setting each factor equal to zero. Since both factors are the same, we only need to solve one equation:3x−5=0 3x - 5 = 0 3x−5=0

STEP 6

Solve for xxx:3x−5=0 3x - 5 = 0 3x−5=0 3x=5 3x = 5 3x=5 x=53 x = \frac{5}{3} x=35​

STEP 7

Determine the solution set. Since both factors were the same, there is only one unique solution:The solution set is {53}\left\{\frac{5}{3}\right\}{35​}. The correct choice is: A. The solution set is {53}\left\{\frac{5}{3}\right\}{35​}.

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1. The equation $9x^2 + 25 = 30x$ is a quadratic equation.
2. Solving the equation involves rearranging it into standard quadratic form.
3. The quadratic can be solved by factoring.

1. Rearrange the equation into standard quadratic form.
2. Factor the quadratic expression.
3. Solve for the roots of the equation.
4. Determine the solution set.

Rearrange the equation to standard quadratic form $ax^2 + bx + c = 0$ . Subtract $30x$ from both sides:
$9x^2 + 25 = 30x$ $9x^2 - 30x + 25 = 0$

Factor the quadratic expression. Look for two numbers that multiply to $9 \times 25 = 225$ and add to $-30$ . The numbers are $-15$ and $-15$ .
$9x^2 - 30x + 25 = (3x - 5)(3x - 5)$

Solve for the roots by setting each factor equal to zero. Since both factors are the same, we only need to solve one equation:
$3x - 5 = 0$

Solve for $x$ :
$3x - 5 = 0$ $3x = 5$ $x = \frac{5}{3}$

Determine the solution set. Since both factors were the same, there is only one unique solution:
The solution set is $\left\{\frac{5}{3}\right\}$ .
The correct choice is: A. The solution set is $\left\{\frac{5}{3}\right\}$ .