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Chapter 9: Problem 19

Solve each equation. $$7 p^{2}+10=26$$

Short Answer

Expert verified
p = \pm \frac{4\sqrt{7}}{7}

Step by step solution

01

Isolate the Quadratic Term

First, subtract 10 from both sides of the equation to isolate the quadratic term. \[ 7p^2 + 10 - 10 = 26 - 10 \] This simplifies to: \[ 7p^2 = 16 \]
02

Solve for the Squared Term

Next, divide both sides of the equation by 7 to solve for \(p^2\). \[ \frac{7p^2}{7} = \frac{16}{7} \] This simplifies to: \[ p^2 = \frac{16}{7} \]
03

Take the Square Root of Both Sides

To solve for \(p\), take the square root of both sides of the equation. \[ p = \pm \sqrt{\frac{16}{7}} \] This can be further simplified: \[ p = \pm \frac{4}{\sqrt{7}} \]
04

Simplify the Square Root Fraction

Rationalize the denominator by multiplying both the numerator and the denominator by \(\sqrt{7}\). \[ p = \pm \frac{4}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}} = \pm \frac{4\sqrt{7}}{7} \]

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Key Concepts

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

quadratic term
A quadratic term is a term in a polynomial that is raised to the power of 2. In the equation given: $$7 p^{2}+10 = 26$$, our quadratic term is $$7p^2$$. To isolate the quadratic term, we need to eliminate any constants on the same side of the equation. We do this by performing basic arithmetic operations.
taking square roots
Once the quadratic term is isolated, we need to solve for the variable by removing the square. We do this by taking the square root of both sides of the equation. Here, we have: $$p^2 = \frac{16}{7}$$, and by taking the square root: $$p = \pm \frac{4}{\sqrt{7}}$$.It's essential to include both the positive and negative solutions, since the square of both a positive and a negative number equals the same positive number.
rationalizing denominators
The result from taking the square root often contains a fraction with an irrational number in the denominator. We rationalize the denominator to simplify the expression and make it more acceptable in mathematical writing. For $$p = \frac{4}{\sqrt{7}}$$, we multiply both the numerator and the denominator by \(\sqrt{7}\) to get: $$p = \frac{4\sqrt{7}}{7}$$. This process helps us convert the fraction into a more standard form without irrational numbers in the denominator.

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Most popular questions from this chapter

Graph each function using a horizontal shift. $$f(x)=(x+5)^{2}$$

Graph each function using transformations. $$f(x)=(x+4)^{2}+2$$

(a) graph the quadratic functions on the same rectangular coordinate system and (b) describe what effect adding a constant, \(h\), inside the parentheses has \(f(x)=x^{2}, g(x)=(x+4)^{2}\) and \(h(x)=(x-4)^{2}\)

(a) rewrite each function in \(f(x)=a(x-h)^{2}+k\) form and (b) graph it using properties. $$f(x)=3 x^{2}-12 x+7$$

(a) solve graphically and (b) write the solution in interval notation. $$x^{2}+6 x+5>0$$
See all solutions

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