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Chapter 9: Problem 46

Solve each equation. $$64 x^{2}+144 x+81=25$$

Short Answer

Expert verified
The solutions are \( x = -\frac{1}{2} \) and \( x = -\frac{7}{4} \).

Step by step solution

01

Move all terms to one side of the equation

First, subtract 25 from both sides of the equation to move all the terms to one side:\[ 64x^{2} + 144x + 81 - 25 = 0 \]This simplifies to:\[ 64x^{2} + 144x + 56 = 0 \]
02

Simplify the equation

Next, divide all terms by the greatest common divisor of the coefficients (8 in this case) to simplify the equation:\[ \frac{64x^{2}}{8} + \frac{144x}{8} + \frac{56}{8} = 0 \]which simplifies to:\[ 8x^{2} + 18x + 7 = 0 \]
03

Use the quadratic formula

Apply the quadratic formula \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \], where \(a = 8\), \(b = 18\), and \(c = 7\).First, compute the discriminant:\[ b^{2} - 4ac = 18^{2} - 4(8)(7) \]\[ = 324 - 224 \]\[ = 100 \]Then, find the roots using the quadratic formula:\[ x = \frac{-18 \pm \sqrt{100}}{2(8)} \]\[ = \frac{-18 \pm 10}{16} \]
04

Simplify the roots

Now separate the two possible solutions:\[ x = \frac{-18 + 10}{16} = \frac{-8}{16} = -\frac{1}{2} \]and\[ x = \frac{-18 - 10}{16} = \frac{-28}{16} = -\frac{7}{4} \]

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Key Concepts

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

quadratic formula
To solve quadratic equations, the quadratic formula is a powerful tool. A quadratic equation takes the form \[ ax^{2} + bx + c = 0 \]. The quadratic formula, which finds the solutions for x, is \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]. Here, * \textbf{a} is the coefficient of \(x^{2}\)* \textbf{b} is the coefficient of x* \textbf{c} is the constant termAlways start by identifying these coefficients from your equation and substituting them into the formula. Don't forget to calculate the discriminant first, as it plays an essential part in finding the roots.
discriminant
The discriminant is a critical part of the quadratic formula and helps determine the nature of the roots. It is calculated as \( b^{2} - 4ac \). Depending on its value, we can infer the following about the roots:
* If the discriminant is positive, there are two distinct real roots.
* If it is zero, there is one real root (repeated root).
* If it is negative, there are no real roots, but rather two complex roots.
In our exercise: \( b^{2} - 4ac = 100 \), because it鈥檚 positive, we know our equation has two distinct real roots.
simplification
Simplification is essential for making equations easier to solve.
In our exercise, we simplify the problem in several ways:
* First, we move all terms to one side of the equation: \[64x^{2}+144x+81-25=0 \].
* Then, we combine like terms to get: \[64x^{2}+144x+56=0 \].
* Finally, we divide by the greatest common divisor of the coefficients, which is 8: \[8x^{2}+18x+7=0 \].
Doing this makes the quadratic formula application simpler, reducing the possibility of making errors in calculations.

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Most popular questions from this chapter

Graph each function using transformations. $$f(x)=(x-4)^{2}-3$$

Graph each function. $$f(x)=-4 x^{2}$$

(a) rewrite each function in \(f(x)=a(x-h)^{2}+k\) form and (b) graph it using properties. $$f(x)=-2 x^{2}-4 x-5$$

Rewrite each function in the \(f(x)=a(x-h)^{2}+k\) form by completing the square. $$f(x)=2 x^{2}-12 x+7$$

(a) rewrite each function in \(f(x)=a(x-h)^{2}+k\) form and (b) graph it by using transformations. $$f(x)=3 x^{2}-6 x-1$$
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