Problem 45 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ 7 x^{2}=4 $$

Step-by-Step Solution

Answer

x = \pm \frac{2 \sqrt{7}}{7}

Step 1: Isolate the quadratic term

To isolate the term with the variable squared, divide both sides of the equation by 7: $$\frac{7 x^{2}}{7} = \frac{4}{7}$$ which simplifies to $$x^{2} = \frac{4}{7}$$.

Step 2: Apply the square root property

To solve for x, apply the square root property to both sides of the equation. Remember to include both the positive and negative square roots: $$x = \pm \sqrt{ \frac{4}{7} }$$.

Step 3: Simplify the radical

Simplify the square root \sqrt{ \frac{4}{7} } by splitting the fraction under the square root into a product of square roots: $$x = \pm \frac{ \sqrt{4} }{ \sqrt{7} }$$. Simplifies further to: $$x = \pm \frac{2}{ \sqrt{7} }$$.

Step 4: Rationalize the denominator

Rationalize the denominator by multiplying both numerator and denominator by \sqrt{7}: $$x = \pm \frac{2 \sqrt{7}}{ \sqrt{7} \sqrt{7} } = \pm \frac{2 \sqrt{7}}{7}.$$

Key Concepts

quadratic equations

Quadratic equations are equations of the form \(ax^{2} + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. In this specific problem, we are dealing with a quadratic equation of the form \(7x^{2} = 4\). The key to solving such equations using the square root property is to isolate the term with the squared variable. Declutter the equation by dividing both sides by the coefficient of \(x^{2}\), which in this case is 7, giving us \(x^{2} = \frac{4}{7}\). This makes it simpler to take the square root on both sides.

simplifying radicals

Simplifying radicals involves breaking down the expression under the square root into more manageable parts. When dealing with \(\frac{4}{7}\), you can simplify the square root by separating the numerator and the denominator under different square roots: \(x = \frac{\text{positive/negative square root of 4}}{\text{square root of 7}}\).
\(\text{The square root of 4}\) simplifies to 2, as \(\text{2 * 2 = 4}\), so now we have \( x = \pm \frac{2}{\text{square root of 7}} \).
Simplifying doesn't always yield a whole number, but breaking the expression up can often make the solution clearer.

rationalizing the denominator

When you have a square root in the denominator, it's common to rationalize it to make the expression easier to understand and work with. Rationalizing involves removing the square root from the denominator by multiplying both the top and bottom of the fraction by the square root present in the denominator. For \(\frac{2}{\text{square root of 7}}\), multiply by \(\text{square root of 7}\) to get

\( \frac{2 * \text{square root of 7}}{\text{square root of 7} * \text{square root of 7}} = \frac{2\text{square root of 7}}{7} \).
This now gives us a rationalized denominator. The solution is expressed as:

$$x = \pm \frac{2\sqrt{7}}{7}$$, making it easier to understand and utilize in further calculations.