Solved on Dec 09, 2023

Solve the quadratic equation $4x^2 - 27 = 1756$ using the square root property. Provide the exact and approximate answers.
Exact answers: $x = \frac{\sqrt{1783}}{2}, -\frac{\sqrt{1783}}{2}$ Approximate answers: $x \approx 23.83, -23.83$

STEP 1

Assumptions
1. We are given the equation $4x^2 - 27 = 1756$ .
2. We need to solve for $x$ using the square root property.
3. We will provide both the exact and approximate solutions for $x$ .
4. The square root property states that if $x^2 = a$ , then $x = \sqrt{a}$ or $x = -\sqrt{a}$ .

STEP 2

First, we want to isolate the $x^2$ term on one side of the equation. To do this, we will add $27$ to both sides of the equation to move the constant term to the right side.
$4x^2 - 27 + 27 = 1756 + 27$

STEP 3

Simplify the equation by combining like terms.
$4x^2 = 1783$

STEP 4

Next, we will use the square root property to solve for $x$ . Before applying the square root property, we need to isolate $x^2$ by dividing both sides of the equation by $4$ .
$\frac{4x^2}{4} = \frac{1783}{4}$

STEP 5

Simplify the equation to get $x^2$ by itself.
$x^2 = \frac{1783}{4}$

STEP 6

Now, apply the square root property to solve for $x$ . According to the square root property, if $x^2 = a$ , then $x = \sqrt{a}$ or $x = -\sqrt{a}$ . Here, $a = \frac{1783}{4}$ .
$x = \sqrt{\frac{1783}{4}} \quad \text{or} \quad x = -\sqrt{\frac{1783}{4}}$

STEP 7

Simplify the square root by taking the square root of the numerator and the denominator separately.
$x = \frac{\sqrt{1783}}{\sqrt{4}} \quad \text{or} \quad x = -\frac{\sqrt{1783}}{\sqrt{4}}$

STEP 8

Since $\sqrt{4} = 2$ , we can further simplify the expression.
$x = \frac{\sqrt{1783}}{2} \quad \text{or} \quad x = -\frac{\sqrt{1783}}{2}$

STEP 9

These are the exact solutions for $x$ . Now we will find the approximate solutions by calculating the numerical value of $\sqrt{1783}$ .

STEP 10

Using a calculator, find the square root of $1783$ .
$\sqrt{1783} \approx 42.24$

STEP 11

Now, substitute the approximate value of $\sqrt{1783}$ into the expressions for $x$ .
$x \approx \frac{42.24}{2} \quad \text{or} \quad x \approx -\frac{42.24}{2}$

STEP 12

Calculate the approximate values for $x$ .
$x \approx 21.12 \quad \text{or} \quad x \approx -21.12$
The exact answers are: $x = \frac{\sqrt{1783}}{2}, -\frac{\sqrt{1783}}{2}$
The approximate answers are: $x \approx 21.12, -21.12$

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Solve the quadratic equation $4x^2 - 27 = 1756$ using the square root property. Provide the exact and approximate answers.
Exact answers: $x = \frac{\sqrt{1783}}{2}, -\frac{\sqrt{1783}}{2}$ Approximate answers: $x \approx 23.83, -23.83$

STEP 1

Assumptions
1. We are given the equation $4x^2 - 27 = 1756$ .
2. We need to solve for $x$ using the square root property.
3. We will provide both the exact and approximate solutions for $x$ .
4. The square root property states that if $x^2 = a$ , then $x = \sqrt{a}$ or $x = -\sqrt{a}$ .

STEP 2

First, we want to isolate the $x^2$ term on one side of the equation. To do this, we will add $27$ to both sides of the equation to move the constant term to the right side.
$4x^2 - 27 + 27 = 1756 + 27$

STEP 3

Simplify the equation by combining like terms.
$4x^2 = 1783$

STEP 4

Next, we will use the square root property to solve for $x$ . Before applying the square root property, we need to isolate $x^2$ by dividing both sides of the equation by $4$ .
$\frac{4x^2}{4} = \frac{1783}{4}$

STEP 5

Simplify the equation to get $x^2$ by itself.
$x^2 = \frac{1783}{4}$

STEP 6

Now, apply the square root property to solve for $x$ . According to the square root property, if $x^2 = a$ , then $x = \sqrt{a}$ or $x = -\sqrt{a}$ . Here, $a = \frac{1783}{4}$ .
$x = \sqrt{\frac{1783}{4}} \quad \text{or} \quad x = -\sqrt{\frac{1783}{4}}$

STEP 7

Simplify the square root by taking the square root of the numerator and the denominator separately.
$x = \frac{\sqrt{1783}}{\sqrt{4}} \quad \text{or} \quad x = -\frac{\sqrt{1783}}{\sqrt{4}}$

STEP 8

Since $\sqrt{4} = 2$ , we can further simplify the expression.
$x = \frac{\sqrt{1783}}{2} \quad \text{or} \quad x = -\frac{\sqrt{1783}}{2}$

STEP 9

These are the exact solutions for $x$ . Now we will find the approximate solutions by calculating the numerical value of $\sqrt{1783}$ .

STEP 10

Using a calculator, find the square root of $1783$ .
$\sqrt{1783} \approx 42.24$

STEP 11

Now, substitute the approximate value of $\sqrt{1783}$ into the expressions for $x$ .
$x \approx \frac{42.24}{2} \quad \text{or} \quad x \approx -\frac{42.24}{2}$

STEP 12

Calculate the approximate values for $x$ .
$x \approx 21.12 \quad \text{or} \quad x \approx -21.12$
The exact answers are: $x = \frac{\sqrt{1783}}{2}, -\frac{\sqrt{1783}}{2}$
The approximate answers are: $x \approx 21.12, -21.12$

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STEP 1

STEP 2

First, we want to isolate the x2x^2x2 term on one side of the equation. To do this, we will add 272727 to both sides of the equation to move the constant term to the right side.4x2−27+27=1756+274x^2 - 27 + 27 = 1756 + 274x2−27+27=1756+27

STEP 3

Simplify the equation by combining like terms.4x2=17834x^2 = 17834x2=1783

STEP 4

Next, we will use the square root property to solve for xxx. Before applying the square root property, we need to isolate x2x^2x2 by dividing both sides of the equation by 444.4x24=17834\frac{4x^2}{4} = \frac{1783}{4}44x2​=41783​

STEP 5

Simplify the equation to get x2x^2x2 by itself.x2=17834x^2 = \frac{1783}{4}x2=41783​

STEP 6

STEP 7

Simplify the square root by taking the square root of the numerator and the denominator separately.x=17834orx=−17834x = \frac{\sqrt{1783}}{\sqrt{4}} \quad \text{or} \quad x = -\frac{\sqrt{1783}}{\sqrt{4}}x=4​1783​​orx=−4​1783​​

STEP 8

Since 4=2\sqrt{4} = 24​=2, we can further simplify the expression.x=17832orx=−17832x = \frac{\sqrt{1783}}{2} \quad \text{or} \quad x = -\frac{\sqrt{1783}}{2}x=21783​​orx=−21783​​

STEP 9

These are the exact solutions for xxx. Now we will find the approximate solutions by calculating the numerical value of 1783\sqrt{1783}1783​.

STEP 10

Using a calculator, find the square root of 178317831783.1783≈42.24\sqrt{1783} \approx 42.241783​≈42.24

STEP 11

Now, substitute the approximate value of 1783\sqrt{1783}1783​ into the expressions for xxx.x≈42.242orx≈−42.242x \approx \frac{42.24}{2} \quad \text{or} \quad x \approx -\frac{42.24}{2}x≈242.24​orx≈−242.24​

STEP 12

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First, we want to isolate the $x^2$ term on one side of the equation. To do this, we will add $27$ to both sides of the equation to move the constant term to the right side.
$4x^2 - 27 + 27 = 1756 + 27$

Simplify the equation by combining like terms.
$4x^2 = 1783$

Next, we will use the square root property to solve for $x$ . Before applying the square root property, we need to isolate $x^2$ by dividing both sides of the equation by $4$ .
$\frac{4x^2}{4} = \frac{1783}{4}$

Simplify the equation to get $x^2$ by itself.
$x^2 = \frac{1783}{4}$

Simplify the square root by taking the square root of the numerator and the denominator separately.
$x = \frac{\sqrt{1783}}{\sqrt{4}} \quad \text{or} \quad x = -\frac{\sqrt{1783}}{\sqrt{4}}$

Since $\sqrt{4} = 2$ , we can further simplify the expression.
$x = \frac{\sqrt{1783}}{2} \quad \text{or} \quad x = -\frac{\sqrt{1783}}{2}$

These are the exact solutions for $x$ . Now we will find the approximate solutions by calculating the numerical value of $\sqrt{1783}$ .

Using a calculator, find the square root of $1783$ .
$\sqrt{1783} \approx 42.24$

Now, substitute the approximate value of $\sqrt{1783}$ into the expressions for $x$ .
$x \approx \frac{42.24}{2} \quad \text{or} \quad x \approx -\frac{42.24}{2}$