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Chapter 8: Problem 25

Use the square root property to solve each equation. \(3 x^{2}-72=0\)

Short Answer

Expert verified
x = \pm 2 \sqrt{6}

Step by step solution

01

- Isolate the quadratic term

Start by isolating the term with the quadratic power. To do this, add 72 to both sides of the equation: 3x^2 - 72 + 72 = 0 + 72 This simplifies to: 3x^2 = 72
02

- Divide by the coefficient of the quadratic term

Next, divide both sides by 3 to solve for x^2: \[ \frac{3x^2}{3} = \frac{72}{3} \] This simplifies to: x^2 = 24
03

- Apply the square root property

Use the square root property, which states if \(a^2 = b\), then \(a = \pm \sqrt{b}\) Apply this property to solve for x: \[ x = \pm \sqrt{24} \] Simplify the square root of 24: \[ x = \pm \sqrt{4 \cdot 6} = \pm 2\sqrt{6} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Root Property
The square root property is a powerful tool for solving quadratic equations. It states that if you have an equation of the form \(a^2 = b\), then the solutions for \(a\) are \(\a = \pm \sqrt{b}\). This is very useful when you have isolated the quadratic term and allows you to solve for the variable easily by taking the square root of both sides.

To apply this property, follow these steps:
  • Ensure the equation is in the form \(a^2 = b\).
  • Take the square root of both sides, remembering to include the \( \pm \) symbol to account for both the positive and negative square roots.
  • Simplify the square root, if possible.
In our example, after isolating \(x^2\) term, we get \(x^2 = 24\). Using the square root property, we find \(x = \pm \sqrt{24}\).
Isolating Quadratic Term
Before you can use the square root property, you must first isolate the quadratic term. This involves getting the term with the variable squared (\(x^2\)) by itself on one side of the equation.

Here鈥檚 how you can do this:
  • Move all other terms to the opposite side using addition or subtraction.
  • If there is a coefficient (number in front of \(x^2\)), divide by that coefficient to simplify.
In our example, we start with the equation \(3x^2 - 72 = 0\).

  • We first add 72 to both sides: \(3x^2 - 72 + 72 = 0 + 72\), giving us \(3x^2 = 72\).
  • Next, we divide by 3: \(x^2 = \frac{72}{3}\), simplifying to \(x^2 = 24\).
Now, the quadratic term \(x^2\) is isolated and we are ready to apply the square root property.
Simplifying Square Roots
Once you have applied the square root property, you may need to simplify the resulting square root. Simplifying square roots involves breaking the number under the square root into its prime factors and simplifying as much as possible.

For example, when you get \(x = \pm \sqrt{24}\), you should:
  • Factor 24 into its prime factors: \(24 = 2 \cdot 12 = 2 \cdot 2 \cdot 6 = 2 \cdot 2 \cdot 2 \cdot 3\).
  • Group any pairs of primes under the square root: \(\sqrt{24} = \sqrt{2^2 \cdot 6} = 2 \sqrt{6}\).
  • So, \(\sqrt{24}\) simplifies to \(2 \sqrt{6}\).

Therefore, the simplified solutions are \(x = \pm 2 \sqrt{6}\). Always ensure the square root is in its simplest form for the final answer.

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Most popular questions from this chapter

In the 1939 classic movie The Wizard of Oz, Ray Bolger's character, the Scarecrow, wants a brain. When the Wizard grants him his "Th.D." (Doctor of Thinkology), the Scarecrow replies with the following statement. Scarecrow: The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side. His statement sounds like the formula for the Pythagorean theorem. In the Scarecrow's statement, he refers to square roots. In applying the formula for the Pythagorean theorem, do we find square roots of the sides? If not, what do we find?

In the 1939 classic movie The Wizard of Oz, Ray Bolger's character, the Scarecrow, wants a brain. When the Wizard grants him his "Th.D." (Doctor of Thinkology), the Scarecrow replies with the following statement. Scarecrow: The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side. His statement sounds like the formula for the Pythagorean theorem. To what kind of triangle does the Scarecrow refer in his statement? To what kind of triangle does the Pythagorean theorem actually refer?

Solve each equation. Check the solutions. $$4 x^{4 / 3}-13 x^{2 / 3}+9=0$$

A model rocket is projected vertically upward from the ground. Its distance s in feet above the ground after t seconds is given by the quadratic function $$ s(t)=-16 t^{2}+256 t $$ to see how quadratic equations and inequalities are related. At what times will the rocket be at ground level? (Hint: Let \(s(t)=0\) and solve the quadratic equation.)

Solve each problem. When appropriate, round answers to the nearest tenth. A toy rocket is launched from ground level. Its distance in feet from the ground in \(t\) seconds is given by $$ s(t)=-16 t^{2}+208 t $$ At what times will the rocket be \(550 \mathrm{ft}\) from the ground?
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