Question Solve the quadratic equation $16(x^2 - 3) - 17 = -16$ for real values of $x$ .

Studdy Solution

STEP 1

Assumptions
1. The given equation is $16(x^2 - 3) - 17 = -16$ .
2. We will solve for $x$ by performing algebraic operations.

STEP 2

First, we need to simplify the equation by distributing the 16 into the parentheses.
$16(x^2 - 3) - 17 = -16$

STEP 3

Distribute the 16 to both terms inside the parentheses.
$16x^2 - 48 - 17 = -16$

STEP 4

Combine the constant terms on the left side of the equation.
$16x^2 - 65 = -16$

STEP 5

Add 65 to both sides of the equation to isolate the term with the variable $x$ on one side.
$16x^2 - 65 + 65 = -16 + 65$

STEP 6

Simplify both sides of the equation after adding 65.
$16x^2 = 49$

STEP 7

Divide both sides of the equation by 16 to solve for $x^2$ .
$x^2 = \frac{49}{16}$

STEP 8

Take the square root of both sides to solve for $x$ . Remember that taking the square root gives us both a positive and negative solution.
$x = \pm \sqrt{\frac{49}{16}}$

STEP 9

Simplify the square root.
$x = \pm \frac{\sqrt{49}}{\sqrt{16}}$

STEP 10

Calculate the square roots.
$x = \pm \frac{7}{4}$

STEP 11

We have two solutions for $x$ .
$x = \frac{7}{4} \quad \text{or} \quad x = -\frac{7}{4}$
The solutions for the equation $16(x^2 - 3) - 17 = -16$ are $x = \frac{7}{4}$ and $x = -\frac{7}{4}$ .

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Question Solve the quadratic equation $16(x^2 - 3) - 17 = -16$ for real values of $x$ .

Studdy Solution

STEP 1

Assumptions
1. The given equation is $16(x^2 - 3) - 17 = -16$ .
2. We will solve for $x$ by performing algebraic operations.

STEP 2

First, we need to simplify the equation by distributing the 16 into the parentheses.
$16(x^2 - 3) - 17 = -16$

STEP 3

Distribute the 16 to both terms inside the parentheses.
$16x^2 - 48 - 17 = -16$

STEP 4

Combine the constant terms on the left side of the equation.
$16x^2 - 65 = -16$

STEP 5

Add 65 to both sides of the equation to isolate the term with the variable $x$ on one side.
$16x^2 - 65 + 65 = -16 + 65$

STEP 6

Simplify both sides of the equation after adding 65.
$16x^2 = 49$

STEP 7

Divide both sides of the equation by 16 to solve for $x^2$ .
$x^2 = \frac{49}{16}$

STEP 8

Take the square root of both sides to solve for $x$ . Remember that taking the square root gives us both a positive and negative solution.
$x = \pm \sqrt{\frac{49}{16}}$

STEP 9

Simplify the square root.
$x = \pm \frac{\sqrt{49}}{\sqrt{16}}$

STEP 10

Calculate the square roots.
$x = \pm \frac{7}{4}$

STEP 11

We have two solutions for $x$ .
$x = \frac{7}{4} \quad \text{or} \quad x = -\frac{7}{4}$
The solutions for the equation $16(x^2 - 3) - 17 = -16$ are $x = \frac{7}{4}$ and $x = -\frac{7}{4}$ .

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Question Solve the quadratic equation 16(x2−3)−17=−1616(x^2 - 3) - 17 = -1616(x2−3)−17=−16 for real values of xxx.

Studdy Solution

STEP 1

Assumptions1. The given equation is 16(x2−3)−17=−16 16(x^2 - 3) - 17 = -16 16(x2−3)−17=−16.2. We will solve for x x x by performing algebraic operations.

STEP 2

First, we need to simplify the equation by distributing the 16 into the parentheses.16(x2−3)−17=−16 16(x^2 - 3) - 17 = -16 16(x2−3)−17=−16

STEP 3

Distribute the 16 to both terms inside the parentheses.16x2−48−17=−16 16x^2 - 48 - 17 = -16 16x2−48−17=−16

STEP 4

Combine the constant terms on the left side of the equation.16x2−65=−16 16x^2 - 65 = -16 16x2−65=−16

STEP 5

Add 65 to both sides of the equation to isolate the term with the variable x x x on one side.16x2−65+65=−16+65 16x^2 - 65 + 65 = -16 + 65 16x2−65+65=−16+65

STEP 6

Simplify both sides of the equation after adding 65.16x2=49 16x^2 = 49 16x2=49

STEP 7

Divide both sides of the equation by 16 to solve for x2 x^2 x2.x2=4916 x^2 = \frac{49}{16} x2=1649​

STEP 8

Take the square root of both sides to solve for x x x. Remember that taking the square root gives us both a positive and negative solution.x=±4916 x = \pm \sqrt{\frac{49}{16}} x=±1649​​

STEP 9

Simplify the square root.x=±4916 x = \pm \frac{\sqrt{49}}{\sqrt{16}} x=±16​49​​

STEP 10

Calculate the square roots.x=±74 x = \pm \frac{7}{4} x=±47​

STEP 11

We have two solutions for x x x.x=74orx=−74 x = \frac{7}{4} \quad \text{or} \quad x = -\frac{7}{4} x=47​orx=−47​The solutions for the equation 16(x2−3)−17=−16 16(x^2 - 3) - 17 = -16 16(x2−3)−17=−16 are x=74 x = \frac{7}{4} x=47​ and x=−74 x = -\frac{7}{4} x=−47​.

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Question Solve the quadratic equation $16(x^2 - 3) - 17 = -16$ for real values of $x$ .

Assumptions
1. The given equation is $16(x^2 - 3) - 17 = -16$ .
2. We will solve for $x$ by performing algebraic operations.

First, we need to simplify the equation by distributing the 16 into the parentheses.
$16(x^2 - 3) - 17 = -16$

Distribute the 16 to both terms inside the parentheses.
$16x^2 - 48 - 17 = -16$

Combine the constant terms on the left side of the equation.
$16x^2 - 65 = -16$

Add 65 to both sides of the equation to isolate the term with the variable $x$ on one side.
$16x^2 - 65 + 65 = -16 + 65$

Simplify both sides of the equation after adding 65.
$16x^2 = 49$

Divide both sides of the equation by 16 to solve for $x^2$ .
$x^2 = \frac{49}{16}$

Take the square root of both sides to solve for $x$ . Remember that taking the square root gives us both a positive and negative solution.
$x = \pm \sqrt{\frac{49}{16}}$

Simplify the square root.
$x = \pm \frac{\sqrt{49}}{\sqrt{16}}$

Calculate the square roots.
$x = \pm \frac{7}{4}$

We have two solutions for $x$ .
$x = \frac{7}{4} \quad \text{or} \quad x = -\frac{7}{4}$
The solutions for the equation $16(x^2 - 3) - 17 = -16$ are $x = \frac{7}{4}$ and $x = -\frac{7}{4}$ .