Math / Algebra

QuestionGiven the following rational expression, determine the values of $x$ for which the expression is undefined. $\frac{6 x^{2}+7 x+2}{6 x^{2}-5 x-6}$
Select one or more: a. 2 b. $\frac{-1}{2}$ C. -2 d. $\frac{-3}{2}$ e. -3 f. $\frac{-2}{3}$ g. $\frac{2}{3}$ h. $\frac{3}{2}$

Studdy Solution

STEP 1

Assumptions
1. The given rational expression is $\frac{6x^2 + 7x + 2}{6x^2 - 5x - 6}$ .
2. The expression is undefined when the denominator is equal to zero.
3. We need to find the values of $x$ that make the denominator zero.

STEP 2

Set the denominator equal to zero to find the values of $x$ that make the expression undefined.
$6x^2 - 5x - 6 = 0$

STEP 3

Solve the quadratic equation $6x^2 - 5x - 6 = 0$ . We can use the quadratic formula:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
where $a = 6$ , $b = -5$ , and $c = -6$ .

STEP 4

Calculate the discriminant $\Delta$ of the quadratic equation.
$\Delta = b^2 - 4ac$

STEP 5

Substitute the values of $a$ , $b$ , and $c$ into the discriminant formula.
$\Delta = (-5)^2 - 4 \cdot 6 \cdot (-6)$

STEP 6

Simplify the expression inside the discriminant.
$\Delta = 25 + 144$

STEP 7

Calculate the value of the discriminant.
$\Delta = 169$

STEP 8

Since the discriminant is positive, there are two distinct real roots. Use the quadratic formula to find the roots.
$x = \frac{-(-5) \pm \sqrt{169}}{2 \cdot 6}$

STEP 9

Simplify the expression inside the quadratic formula.
$x = \frac{5 \pm 13}{12}$

STEP 10

Calculate the two possible values for $x$ .
For the positive root:
$x_1 = \frac{5 + 13}{12} = \frac{18}{12} = \frac{3}{2}$

STEP 11

For the negative root:
$x_2 = \frac{5 - 13}{12} = \frac{-8}{12} = \frac{-2}{3}$

STEP 12

List the values of $x$ that make the denominator zero and thus the expression undefined.
The values of $x$ are $\frac{3}{2}$ and $\frac{-2}{3}$ .

STEP 13

Identify the correct options from the given choices.
The correct options are: g. $\frac{-2}{3}$ h. $\frac{3}{2}$

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QuestionGiven the following rational expression, determine the values of $x$ for which the expression is undefined. $\frac{6 x^{2}+7 x+2}{6 x^{2}-5 x-6}$
Select one or more: a. 2 b. $\frac{-1}{2}$ C. -2 d. $\frac{-3}{2}$ e. -3 f. $\frac{-2}{3}$ g. $\frac{2}{3}$ h. $\frac{3}{2}$

Studdy Solution

STEP 1

Assumptions
1. The given rational expression is $\frac{6x^2 + 7x + 2}{6x^2 - 5x - 6}$ .
2. The expression is undefined when the denominator is equal to zero.
3. We need to find the values of $x$ that make the denominator zero.

STEP 2

Set the denominator equal to zero to find the values of $x$ that make the expression undefined.
$6x^2 - 5x - 6 = 0$

STEP 3

Solve the quadratic equation $6x^2 - 5x - 6 = 0$ . We can use the quadratic formula:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
where $a = 6$ , $b = -5$ , and $c = -6$ .

STEP 4

Calculate the discriminant $\Delta$ of the quadratic equation.
$\Delta = b^2 - 4ac$

STEP 5

Substitute the values of $a$ , $b$ , and $c$ into the discriminant formula.
$\Delta = (-5)^2 - 4 \cdot 6 \cdot (-6)$

STEP 6

Simplify the expression inside the discriminant.
$\Delta = 25 + 144$

STEP 7

Calculate the value of the discriminant.
$\Delta = 169$

STEP 8

Since the discriminant is positive, there are two distinct real roots. Use the quadratic formula to find the roots.
$x = \frac{-(-5) \pm \sqrt{169}}{2 \cdot 6}$

STEP 9

Simplify the expression inside the quadratic formula.
$x = \frac{5 \pm 13}{12}$

STEP 10

Calculate the two possible values for $x$ .
For the positive root:
$x_1 = \frac{5 + 13}{12} = \frac{18}{12} = \frac{3}{2}$

STEP 11

For the negative root:
$x_2 = \frac{5 - 13}{12} = \frac{-8}{12} = \frac{-2}{3}$

STEP 12

List the values of $x$ that make the denominator zero and thus the expression undefined.
The values of $x$ are $\frac{3}{2}$ and $\frac{-2}{3}$ .

STEP 13

Identify the correct options from the given choices.
The correct options are: g. $\frac{-2}{3}$ h. $\frac{3}{2}$

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Studdy Solution

STEP 1

Assumptions1. The given rational expression is 6x2+7x+26x2−5x−6\frac{6x^2 + 7x + 2}{6x^2 - 5x - 6}6x2−5x−66x2+7x+2​.2. The expression is undefined when the denominator is equal to zero.3. We need to find the values of xxx that make the denominator zero.

STEP 2

Set the denominator equal to zero to find the values of xxx that make the expression undefined.6x2−5x−6=06x^2 - 5x - 6 = 06x2−5x−6=0

STEP 3

Solve the quadratic equation 6x2−5x−6=06x^2 - 5x - 6 = 06x2−5x−6=0. We can use the quadratic formula:x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac​​where a=6a = 6a=6, b=−5b = -5b=−5, and c=−6c = -6c=−6.

STEP 4

Calculate the discriminant Δ\DeltaΔ of the quadratic equation.Δ=b2−4ac\Delta = b^2 - 4acΔ=b2−4ac

STEP 5

Substitute the values of aaa, bbb, and ccc into the discriminant formula.Δ=(−5)2−4⋅6⋅(−6)\Delta = (-5)^2 - 4 \cdot 6 \cdot (-6)Δ=(−5)2−4⋅6⋅(−6)

STEP 6

Simplify the expression inside the discriminant.Δ=25+144\Delta = 25 + 144Δ=25+144

STEP 7

Calculate the value of the discriminant.Δ=169\Delta = 169Δ=169

STEP 8

Since the discriminant is positive, there are two distinct real roots. Use the quadratic formula to find the roots.x=−(−5)±1692⋅6x = \frac{-(-5) \pm \sqrt{169}}{2 \cdot 6}x=2⋅6−(−5)±169​​

STEP 9

Simplify the expression inside the quadratic formula.x=5±1312x = \frac{5 \pm 13}{12}x=125±13​

STEP 10

Calculate the two possible values for xxx.For the positive root:x1=5+1312=1812=32x_1 = \frac{5 + 13}{12} = \frac{18}{12} = \frac{3}{2}x1​=125+13​=1218​=23​

STEP 11

For the negative root:x2=5−1312=−812=−23x_2 = \frac{5 - 13}{12} = \frac{-8}{12} = \frac{-2}{3}x2​=125−13​=12−8​=3−2​

STEP 12

List the values of xxx that make the denominator zero and thus the expression undefined.The values of xxx are 32\frac{3}{2}23​ and −23\frac{-2}{3}3−2​.

STEP 13

Identify the correct options from the given choices.The correct options are: g. −23\frac{-2}{3}3−2​ h. 32\frac{3}{2}23​

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Assumptions
1. The given rational expression is $\frac{6x^2 + 7x + 2}{6x^2 - 5x - 6}$ .
2. The expression is undefined when the denominator is equal to zero.
3. We need to find the values of $x$ that make the denominator zero.

Set the denominator equal to zero to find the values of $x$ that make the expression undefined.
$6x^2 - 5x - 6 = 0$

Solve the quadratic equation $6x^2 - 5x - 6 = 0$ . We can use the quadratic formula:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
where $a = 6$ , $b = -5$ , and $c = -6$ .

Calculate the discriminant $\Delta$ of the quadratic equation.
$\Delta = b^2 - 4ac$

Substitute the values of $a$ , $b$ , and $c$ into the discriminant formula.
$\Delta = (-5)^2 - 4 \cdot 6 \cdot (-6)$

Simplify the expression inside the discriminant.
$\Delta = 25 + 144$

Calculate the value of the discriminant.
$\Delta = 169$

Since the discriminant is positive, there are two distinct real roots. Use the quadratic formula to find the roots.
$x = \frac{-(-5) \pm \sqrt{169}}{2 \cdot 6}$

Simplify the expression inside the quadratic formula.
$x = \frac{5 \pm 13}{12}$

Calculate the two possible values for $x$ .
For the positive root:
$x_1 = \frac{5 + 13}{12} = \frac{18}{12} = \frac{3}{2}$

For the negative root:
$x_2 = \frac{5 - 13}{12} = \frac{-8}{12} = \frac{-2}{3}$

List the values of $x$ that make the denominator zero and thus the expression undefined.
The values of $x$ are $\frac{3}{2}$ and $\frac{-2}{3}$ .

Identify the correct options from the given choices.
The correct options are: g. $\frac{-2}{3}$ h. $\frac{3}{2}$