Solved on Dec 06, 2023

Find the discriminant of the quadratic equation $4x^2 - 12x - 6 = 0$ .

STEP 1

Assumptions
1. The quadratic function is given by $4x^2 - 6 = -12x$ .
2. The standard form of a quadratic function is $ax^2 + bx + c$ .
3. The discriminant of a quadratic function in standard form is given by $D = b^2 - 4ac$ .

STEP 2

First, we need to rewrite the given quadratic function in standard form. To do this, we move all terms to one side of the equation to get zero on the other side.
$4x^2 + 12x - 6 = 0$

STEP 3

Now that we have the quadratic function in standard form, we can identify the coefficients $a$ , $b$ , and $c$ .
$a = 4, \quad b = 12, \quad c = -6$

STEP 4

Next, we will use the formula for the discriminant of a quadratic equation to find the discriminant.
$D = b^2 - 4ac$

STEP 5

Substitute the identified values of $a$ , $b$ , and $c$ into the discriminant formula.
$D = (12)^2 - 4(4)(-6)$

STEP 6

Calculate the square of $b$ .
$D = 144 - 4(4)(-6)$

STEP 7

Multiply $4ac$ .
$D = 144 - 4(4)(-6) = 144 - (-96)$

STEP 8

Subtract to find the discriminant.
$D = 144 + 96$

STEP 9

Calculate the final value of the discriminant.
$D = 240$
The discriminant of the quadratic function $4x^2 - 6 = -12x$ is $240$ .

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Find the discriminant of the quadratic equation $4x^2 - 12x - 6 = 0$ .

STEP 1

Assumptions
1. The quadratic function is given by $4x^2 - 6 = -12x$ .
2. The standard form of a quadratic function is $ax^2 + bx + c$ .
3. The discriminant of a quadratic function in standard form is given by $D = b^2 - 4ac$ .

STEP 2

First, we need to rewrite the given quadratic function in standard form. To do this, we move all terms to one side of the equation to get zero on the other side.
$4x^2 + 12x - 6 = 0$

STEP 3

Now that we have the quadratic function in standard form, we can identify the coefficients $a$ , $b$ , and $c$ .
$a = 4, \quad b = 12, \quad c = -6$

STEP 4

Next, we will use the formula for the discriminant of a quadratic equation to find the discriminant.
$D = b^2 - 4ac$

STEP 5

Substitute the identified values of $a$ , $b$ , and $c$ into the discriminant formula.
$D = (12)^2 - 4(4)(-6)$

STEP 6

Calculate the square of $b$ .
$D = 144 - 4(4)(-6)$

STEP 7

Multiply $4ac$ .
$D = 144 - 4(4)(-6) = 144 - (-96)$

STEP 8

Subtract to find the discriminant.
$D = 144 + 96$

STEP 9

Calculate the final value of the discriminant.
$D = 240$
The discriminant of the quadratic function $4x^2 - 6 = -12x$ is $240$ .

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Find the discriminant of the quadratic equation 4x2−12x−6=04x^2 - 12x - 6 = 04x2−12x−6=0.

STEP 1

Assumptions1. The quadratic function is given by 4x2−6=−12x4x^2 - 6 = -12x4x2−6=−12x.2. The standard form of a quadratic function is ax2+bx+cax^2 + bx + cax2+bx+c.3. The discriminant of a quadratic function in standard form is given by D=b2−4acD = b^2 - 4acD=b2−4ac.

STEP 2

First, we need to rewrite the given quadratic function in standard form. To do this, we move all terms to one side of the equation to get zero on the other side.4x2+12x−6=04x^2 + 12x - 6 = 04x2+12x−6=0

STEP 3

Now that we have the quadratic function in standard form, we can identify the coefficients aaa, bbb, and ccc.a=4,b=12,c=−6a = 4, \quad b = 12, \quad c = -6a=4,b=12,c=−6

STEP 4

Next, we will use the formula for the discriminant of a quadratic equation to find the discriminant.D=b2−4acD = b^2 - 4acD=b2−4ac

STEP 5

Substitute the identified values of aaa, bbb, and ccc into the discriminant formula.D=(12)2−4(4)(−6)D = (12)^2 - 4(4)(-6)D=(12)2−4(4)(−6)

STEP 6

Calculate the square of bbb.D=144−4(4)(−6)D = 144 - 4(4)(-6)D=144−4(4)(−6)

STEP 7

Multiply 4ac4ac4ac.D=144−4(4)(−6)=144−(−96)D = 144 - 4(4)(-6) = 144 - (-96)D=144−4(4)(−6)=144−(−96)

STEP 8

Subtract to find the discriminant.D=144+96D = 144 + 96D=144+96

STEP 9

Calculate the final value of the discriminant.D=240D = 240D=240The discriminant of the quadratic function 4x2−6=−12x4x^2 - 6 = -12x4x2−6=−12x is 240240240.

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Find the discriminant of the quadratic equation $4x^2 - 12x - 6 = 0$ .

Assumptions
1. The quadratic function is given by $4x^2 - 6 = -12x$ .
2. The standard form of a quadratic function is $ax^2 + bx + c$ .
3. The discriminant of a quadratic function in standard form is given by $D = b^2 - 4ac$ .

First, we need to rewrite the given quadratic function in standard form. To do this, we move all terms to one side of the equation to get zero on the other side.
$4x^2 + 12x - 6 = 0$

Now that we have the quadratic function in standard form, we can identify the coefficients $a$ , $b$ , and $c$ .
$a = 4, \quad b = 12, \quad c = -6$

Next, we will use the formula for the discriminant of a quadratic equation to find the discriminant.
$D = b^2 - 4ac$

Substitute the identified values of $a$ , $b$ , and $c$ into the discriminant formula.
$D = (12)^2 - 4(4)(-6)$

Calculate the square of $b$ .
$D = 144 - 4(4)(-6)$

Multiply $4ac$ .
$D = 144 - 4(4)(-6) = 144 - (-96)$

Subtract to find the discriminant.
$D = 144 + 96$

Calculate the final value of the discriminant.
$D = 240$
The discriminant of the quadratic function $4x^2 - 6 = -12x$ is $240$ .