Math / Algebra

QuestionQuestion Show Exampl
What are the roots of the equation $9 x^{2}-36 x+37=0$ in simplest $a+b i$ form?
Answer Attempt 1 out of 2. ( Additional Solution $\Theta$ No Solution

Studdy Solution

STEP 1

1. The equation $9x^2 - 36x + 37 = 0$ is a quadratic equation.
2. The roots can be complex, given in the form $a + bi$ .
3. We will use the quadratic formula to find the roots.

STEP 2

1. Identify coefficients for the quadratic formula.
2. Calculate the discriminant.
3. Use the quadratic formula to find the roots.
4. Simplify the roots into the form $a + bi$ .

STEP 3

Identify the coefficients $a$ , $b$ , and $c$ from the quadratic equation $ax^2 + bx + c = 0$ . For the equation $9x^2 - 36x + 37 = 0$ , we have:
$a = 9, \quad b = -36, \quad c = 37$

STEP 4

Calculate the discriminant $\Delta$ using the formula $\Delta = b^2 - 4ac$ .
$\Delta = (-36)^2 - 4 \times 9 \times 37$

STEP 5

Compute the value of the discriminant:
$\Delta = 1296 - 1332$ $\Delta = -36$

STEP 6

Since the discriminant is negative, the roots are complex. Use the quadratic formula $x = \frac{-b \pm \sqrt{\Delta}}{2a}$ .
$x = \frac{-(-36) \pm \sqrt{-36}}{2 \times 9}$

STEP 7

Simplify the expression:
$x = \frac{36 \pm \sqrt{-36}}{18}$

STEP 8

Since $\sqrt{-36} = 6i$ , substitute this into the expression:
$x = \frac{36 \pm 6i}{18}$

STEP 9

Simplify the fraction:
$x = \frac{36}{18} \pm \frac{6i}{18}$ $x = 2 \pm \frac{1}{3}i$

STEP 10

Express the roots in the form $a + bi$ :
The roots are $2 + \frac{1}{3}i$ and $2 - \frac{1}{3}i$ .
The roots of the equation in simplest $a + bi$ form are:
$\boxed{2 \pm \frac{1}{3}i}$

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QuestionQuestion Show Exampl
What are the roots of the equation $9 x^{2}-36 x+37=0$ in simplest $a+b i$ form?
Answer Attempt 1 out of 2. ( Additional Solution $\Theta$ No Solution

Studdy Solution

STEP 1

1. The equation $9x^2 - 36x + 37 = 0$ is a quadratic equation.
2. The roots can be complex, given in the form $a + bi$ .
3. We will use the quadratic formula to find the roots.

STEP 2

1. Identify coefficients for the quadratic formula.
2. Calculate the discriminant.
3. Use the quadratic formula to find the roots.
4. Simplify the roots into the form $a + bi$ .

STEP 3

Identify the coefficients $a$ , $b$ , and $c$ from the quadratic equation $ax^2 + bx + c = 0$ . For the equation $9x^2 - 36x + 37 = 0$ , we have:
$a = 9, \quad b = -36, \quad c = 37$

STEP 4

Calculate the discriminant $\Delta$ using the formula $\Delta = b^2 - 4ac$ .
$\Delta = (-36)^2 - 4 \times 9 \times 37$

STEP 5

Compute the value of the discriminant:
$\Delta = 1296 - 1332$ $\Delta = -36$

STEP 6

Since the discriminant is negative, the roots are complex. Use the quadratic formula $x = \frac{-b \pm \sqrt{\Delta}}{2a}$ .
$x = \frac{-(-36) \pm \sqrt{-36}}{2 \times 9}$

STEP 7

Simplify the expression:
$x = \frac{36 \pm \sqrt{-36}}{18}$

STEP 8

Since $\sqrt{-36} = 6i$ , substitute this into the expression:
$x = \frac{36 \pm 6i}{18}$

STEP 9

Simplify the fraction:
$x = \frac{36}{18} \pm \frac{6i}{18}$ $x = 2 \pm \frac{1}{3}i$

STEP 10

Express the roots in the form $a + bi$ :
The roots are $2 + \frac{1}{3}i$ and $2 - \frac{1}{3}i$ .
The roots of the equation in simplest $a + bi$ form are:
$\boxed{2 \pm \frac{1}{3}i}$

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QuestionQuestion Show ExamplWhat are the roots of the equation 9x2−36x+37=09 x^{2}-36 x+37=09x2−36x+37=0 in simplest a+bia+b ia+bi form?Answer Attempt 1 out of 2. ( Additional Solution Θ\ThetaΘ No Solution

Studdy Solution

STEP 1

1. The equation 9x2−36x+37=0 9x^2 - 36x + 37 = 0 9x2−36x+37=0 is a quadratic equation.2. The roots can be complex, given in the form a+bi a + bi a+bi.3. We will use the quadratic formula to find the roots.

STEP 2

1. Identify coefficients for the quadratic formula.2. Calculate the discriminant.3. Use the quadratic formula to find the roots.4. Simplify the roots into the form a+bi a + bi a+bi.

STEP 3

Identify the coefficients a a a, b b b, and c c c from the quadratic equation ax2+bx+c=0 ax^2 + bx + c = 0 ax2+bx+c=0. For the equation 9x2−36x+37=0 9x^2 - 36x + 37 = 0 9x2−36x+37=0, we have:a=9,b=−36,c=37 a = 9, \quad b = -36, \quad c = 37 a=9,b=−36,c=37

STEP 4

Calculate the discriminant Δ\DeltaΔ using the formula Δ=b2−4ac\Delta = b^2 - 4acΔ=b2−4ac.Δ=(−36)2−4×9×37 \Delta = (-36)^2 - 4 \times 9 \times 37 Δ=(−36)2−4×9×37

STEP 5

Compute the value of the discriminant:Δ=1296−1332 \Delta = 1296 - 1332 Δ=1296−1332 Δ=−36 \Delta = -36 Δ=−36

STEP 6

Since the discriminant is negative, the roots are complex. Use the quadratic formula x=−b±Δ2a x = \frac{-b \pm \sqrt{\Delta}}{2a} x=2a−b±Δ​​.x=−(−36)±−362×9 x = \frac{-(-36) \pm \sqrt{-36}}{2 \times 9} x=2×9−(−36)±−36​​

STEP 7

Simplify the expression:x=36±−3618 x = \frac{36 \pm \sqrt{-36}}{18} x=1836±−36​​

STEP 8

Since −36=6i\sqrt{-36} = 6i−36​=6i, substitute this into the expression:x=36±6i18 x = \frac{36 \pm 6i}{18} x=1836±6i​

STEP 9

Simplify the fraction:x=3618±6i18 x = \frac{36}{18} \pm \frac{6i}{18} x=1836​±186i​ x=2±13i x = 2 \pm \frac{1}{3}i x=2±31​i

STEP 10

Express the roots in the form a+bi a + bi a+bi:The roots are 2+13i 2 + \frac{1}{3}i 2+31​i and 2−13i 2 - \frac{1}{3}i 2−31​i. The roots of the equation in simplest a+bi a + bi a+bi form are:2±13i \boxed{2 \pm \frac{1}{3}i} 2±31​i​

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QuestionQuestion Show Exampl
What are the roots of the equation $9 x^{2}-36 x+37=0$ in simplest $a+b i$ form?
Answer Attempt 1 out of 2. ( Additional Solution $\Theta$ No Solution

1. The equation $9x^2 - 36x + 37 = 0$ is a quadratic equation.
2. The roots can be complex, given in the form $a + bi$ .
3. We will use the quadratic formula to find the roots.

1. Identify coefficients for the quadratic formula.
2. Calculate the discriminant.
3. Use the quadratic formula to find the roots.
4. Simplify the roots into the form $a + bi$ .

Identify the coefficients $a$ , $b$ , and $c$ from the quadratic equation $ax^2 + bx + c = 0$ . For the equation $9x^2 - 36x + 37 = 0$ , we have:
$a = 9, \quad b = -36, \quad c = 37$

Calculate the discriminant $\Delta$ using the formula $\Delta = b^2 - 4ac$ .
$\Delta = (-36)^2 - 4 \times 9 \times 37$

Compute the value of the discriminant:
$\Delta = 1296 - 1332$ $\Delta = -36$

Since the discriminant is negative, the roots are complex. Use the quadratic formula $x = \frac{-b \pm \sqrt{\Delta}}{2a}$ .
$x = \frac{-(-36) \pm \sqrt{-36}}{2 \times 9}$

Simplify the expression:
$x = \frac{36 \pm \sqrt{-36}}{18}$

Since $\sqrt{-36} = 6i$ , substitute this into the expression:
$x = \frac{36 \pm 6i}{18}$

Simplify the fraction:
$x = \frac{36}{18} \pm \frac{6i}{18}$ $x = 2 \pm \frac{1}{3}i$

Express the roots in the form $a + bi$ :
The roots are $2 + \frac{1}{3}i$ and $2 - \frac{1}{3}i$ .
The roots of the equation in simplest $a + bi$ form are:
$\boxed{2 \pm \frac{1}{3}i}$