Solved on Feb 11, 2024

Solve for $w$ in the quadratic equation $4 w^{2} = -11 w - 6$ . If there are multiple solutions, list them separately.

STEP 1

Assumptions
1. We are given a quadratic equation in the form $4w^2 + 11w + 6 = 0$ .
2. We need to solve for the variable $w$ .
3. There may be two solutions since this is a quadratic equation.

STEP 2

Rewrite the given equation in standard quadratic form.
$4w^2 + 11w + 6 = 0$

STEP 3

We will use the quadratic formula to solve for $w$ . The quadratic formula is given by:
$w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
where $a$ , $b$ , and $c$ are the coefficients of the terms $w^2$ , $w$ , and the constant term, respectively, in the quadratic equation $aw^2 + bw + c = 0$ .

STEP 4

Identify the coefficients $a$ , $b$ , and $c$ from the equation $4w^2 + 11w + 6 = 0$ .
$a = 4, \quad b = 11, \quad c = 6$

STEP 5

Substitute the values of $a$ , $b$ , and $c$ into the quadratic formula.
$w = \frac{-11 \pm \sqrt{11^2 - 4 \cdot 4 \cdot 6}}{2 \cdot 4}$

STEP 6

Simplify the expression under the square root (the discriminant).
$11^2 - 4 \cdot 4 \cdot 6 = 121 - 96$

STEP 7

Calculate the discriminant.
$121 - 96 = 25$

STEP 8

Substitute the discriminant back into the quadratic formula.
$w = \frac{-11 \pm \sqrt{25}}{2 \cdot 4}$

STEP 9

Take the square root of the discriminant.
$\sqrt{25} = 5$

STEP 10

Substitute the square root of the discriminant into the quadratic formula.
$w = \frac{-11 \pm 5}{8}$

STEP 11

Solve for $w$ by using the plus sign in the $\pm$ symbol.
$w = \frac{-11 + 5}{8}$

STEP 12

Calculate the value of $w$ for the plus sign.
$w = \frac{-6}{8}$

STEP 13

Simplify the fraction.
$w = -\frac{3}{4}$

STEP 14

Now solve for $w$ by using the minus sign in the $\pm$ symbol.
$w = \frac{-11 - 5}{8}$

STEP 15

Calculate the value of $w$ for the minus sign.
$w = \frac{-16}{8}$

STEP 16

Simplify the fraction.
$w = -2$
The solutions for $w$ are $-\frac{3}{4}$ and $-2$ .

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Solve for $w$ in the quadratic equation $4 w^{2} = -11 w - 6$ . If there are multiple solutions, list them separately.

STEP 1

Assumptions
1. We are given a quadratic equation in the form $4w^2 + 11w + 6 = 0$ .
2. We need to solve for the variable $w$ .
3. There may be two solutions since this is a quadratic equation.

STEP 2

Rewrite the given equation in standard quadratic form.
$4w^2 + 11w + 6 = 0$

STEP 3

We will use the quadratic formula to solve for $w$ . The quadratic formula is given by:
$w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
where $a$ , $b$ , and $c$ are the coefficients of the terms $w^2$ , $w$ , and the constant term, respectively, in the quadratic equation $aw^2 + bw + c = 0$ .

STEP 4

Identify the coefficients $a$ , $b$ , and $c$ from the equation $4w^2 + 11w + 6 = 0$ .
$a = 4, \quad b = 11, \quad c = 6$

STEP 5

Substitute the values of $a$ , $b$ , and $c$ into the quadratic formula.
$w = \frac{-11 \pm \sqrt{11^2 - 4 \cdot 4 \cdot 6}}{2 \cdot 4}$

STEP 6

Simplify the expression under the square root (the discriminant).
$11^2 - 4 \cdot 4 \cdot 6 = 121 - 96$

STEP 7

Calculate the discriminant.
$121 - 96 = 25$

STEP 8

Substitute the discriminant back into the quadratic formula.
$w = \frac{-11 \pm \sqrt{25}}{2 \cdot 4}$

STEP 9

Take the square root of the discriminant.
$\sqrt{25} = 5$

STEP 10

Substitute the square root of the discriminant into the quadratic formula.
$w = \frac{-11 \pm 5}{8}$

STEP 11

Solve for $w$ by using the plus sign in the $\pm$ symbol.
$w = \frac{-11 + 5}{8}$

STEP 12

Calculate the value of $w$ for the plus sign.
$w = \frac{-6}{8}$

STEP 13

Simplify the fraction.
$w = -\frac{3}{4}$

STEP 14

Now solve for $w$ by using the minus sign in the $\pm$ symbol.
$w = \frac{-11 - 5}{8}$

STEP 15

Calculate the value of $w$ for the minus sign.
$w = \frac{-16}{8}$

STEP 16

Simplify the fraction.
$w = -2$
The solutions for $w$ are $-\frac{3}{4}$ and $-2$ .

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Solve for www in the quadratic equation 4w2=−11w−64 w^{2} = -11 w - 64w2=−11w−6. If there are multiple solutions, list them separately.

STEP 1

Assumptions1. We are given a quadratic equation in the form 4w2+11w+6=04w^2 + 11w + 6 = 04w2+11w+6=0.2. We need to solve for the variable www.3. There may be two solutions since this is a quadratic equation.

STEP 2

Rewrite the given equation in standard quadratic form.4w2+11w+6=04w^2 + 11w + 6 = 04w2+11w+6=0

STEP 3

STEP 4

Identify the coefficients aaa, bbb, and ccc from the equation 4w2+11w+6=04w^2 + 11w + 6 = 04w2+11w+6=0.a=4,b=11,c=6a = 4, \quad b = 11, \quad c = 6a=4,b=11,c=6

STEP 5

Substitute the values of aaa, bbb, and ccc into the quadratic formula.w=−11±112−4⋅4⋅62⋅4w = \frac{-11 \pm \sqrt{11^2 - 4 \cdot 4 \cdot 6}}{2 \cdot 4}w=2⋅4−11±112−4⋅4⋅6​​

STEP 6

Simplify the expression under the square root (the discriminant).112−4⋅4⋅6=121−9611^2 - 4 \cdot 4 \cdot 6 = 121 - 96112−4⋅4⋅6=121−96

STEP 7

Calculate the discriminant.121−96=25121 - 96 = 25121−96=25

STEP 8

Substitute the discriminant back into the quadratic formula.w=−11±252⋅4w = \frac{-11 \pm \sqrt{25}}{2 \cdot 4}w=2⋅4−11±25​​

STEP 9

Take the square root of the discriminant.25=5\sqrt{25} = 525​=5

STEP 10

Substitute the square root of the discriminant into the quadratic formula.w=−11±58w = \frac{-11 \pm 5}{8}w=8−11±5​

STEP 11

Solve for www by using the plus sign in the ±\pm± symbol.w=−11+58w = \frac{-11 + 5}{8}w=8−11+5​

STEP 12

Calculate the value of www for the plus sign.w=−68w = \frac{-6}{8}w=8−6​

STEP 13

Simplify the fraction.w=−34w = -\frac{3}{4}w=−43​

STEP 14

Now solve for www by using the minus sign in the ±\pm± symbol.w=−11−58w = \frac{-11 - 5}{8}w=8−11−5​

STEP 15

Calculate the value of www for the minus sign.w=−168w = \frac{-16}{8}w=8−16​

STEP 16

Simplify the fraction.w=−2w = -2w=−2The solutions for www are −34-\frac{3}{4}−43​ and −2-2−2.

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Solve for $w$ in the quadratic equation $4 w^{2} = -11 w - 6$ . If there are multiple solutions, list them separately.

Assumptions
1. We are given a quadratic equation in the form $4w^2 + 11w + 6 = 0$ .
2. We need to solve for the variable $w$ .
3. There may be two solutions since this is a quadratic equation.

Rewrite the given equation in standard quadratic form.
$4w^2 + 11w + 6 = 0$

Identify the coefficients $a$ , $b$ , and $c$ from the equation $4w^2 + 11w + 6 = 0$ .
$a = 4, \quad b = 11, \quad c = 6$

Substitute the values of $a$ , $b$ , and $c$ into the quadratic formula.
$w = \frac{-11 \pm \sqrt{11^2 - 4 \cdot 4 \cdot 6}}{2 \cdot 4}$

Simplify the expression under the square root (the discriminant).
$11^2 - 4 \cdot 4 \cdot 6 = 121 - 96$

Calculate the discriminant.
$121 - 96 = 25$

Substitute the discriminant back into the quadratic formula.
$w = \frac{-11 \pm \sqrt{25}}{2 \cdot 4}$

Take the square root of the discriminant.
$\sqrt{25} = 5$

Substitute the square root of the discriminant into the quadratic formula.
$w = \frac{-11 \pm 5}{8}$

Solve for $w$ by using the plus sign in the $\pm$ symbol.
$w = \frac{-11 + 5}{8}$

Calculate the value of $w$ for the plus sign.
$w = \frac{-6}{8}$

Simplify the fraction.
$w = -\frac{3}{4}$

Now solve for $w$ by using the minus sign in the $\pm$ symbol.
$w = \frac{-11 - 5}{8}$

Calculate the value of $w$ for the minus sign.
$w = \frac{-16}{8}$

Simplify the fraction.
$w = -2$
The solutions for $w$ are $-\frac{3}{4}$ and $-2$ .