/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 53 \(c^{2}+14 c+48=0\)... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 53

\(c^{2}+14 c+48=0\)

Short Answer

Expert verified
The solutions are \( c = -6 \) and \( c = -8 \).

Step by step solution

01

- Identify the quadratic equation

The equation given is a standard quadratic equation, which can be written in the form: \[ ax^2 + bx + c = 0 \]Here, \(a = 1\), \(b = 14\), and \(c = 48\).
02

- Use the quadratic formula

The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Substitute \(a = 1\), \(b = 14\), and \(c = 48\) into the formula.
03

- Calculate the discriminant

Calculate the discriminant (\( \Delta \)) using:\[ \Delta = b^2 - 4ac \]Substitute the values: \[ \Delta = 14^2 - 4(1)(48) = 196 - 192 = 4 \]
04

- Solve for the roots

Using the quadratic formula:\[ c = \frac{-14 \pm \sqrt{4}}{2(1)} = \frac{-14 \pm 2}{2} \]This gives us two solutions:\[ c_1 = \frac{-14 + 2}{2} = -6 \]\[ c_2 = \frac{-14 - 2}{2} = -8 \]
05

- Write the final answer

The solutions to the quadratic equation \( c^2 + 14c + 48 = 0 \) are \( c = -6 \) and \( c = -8 \).

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Key Concepts

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

quadratic formula
The quadratic formula is a powerful tool for solving quadratic equations. Quadratic equations are in the form \[ ax^2 + bx + c = 0 \] where \( a \), \( b \), and \( c \) are constants. The quadratic formula provides solutions for these equations using:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here's a step-by-step on how to use it:
  • Identify the coefficients \( a \), \( b \), and \( c \) from the equation.
  • Substitute these values into the formula.
  • Calculate the discriminant, which is \( b^2 - 4ac \).
  • Solve for \( x \) by simplifying the expression under the square root and completing the operations.

In our example equation \( c^2 + 14c + 48 = 0 \), we identified \( a = 1 \), \( b = 14 \), and \( c = 48 \). We substituted these into the quadratic formula to find our solution.
discriminant
The discriminant is a key part of the quadratic formula. It helps determine the nature of the solutions to the quadratic equation.
The formula for the discriminant is:\[ \Delta = b^2 - 4ac \]
The discriminant reveals several important details about the roots of the equation:
  • If \( \Delta > 0 \), the equation has two distinct real roots.
  • If \( \Delta = 0 \), the equation has exactly one real root (also called a repeated or double root).
  • If \( \Delta < 0 \), the equation has two complex roots.

In the example above, we calculated the discriminant as:\[ \Delta = 14^2 - 4(1)(48) = 196 - 192 = 4 \]Since the discriminant is 4 (greater than 0), we know our equation has two distinct real roots.
solving quadratic equations
Solving quadratic equations means finding the values of \( x \) (or another variable) that make the equation true.
Using our example equation, \( c^2 + 14c + 48 = 0 \), and the quadratic formula, we can find these values step by step:
  • Identify the coefficients: \( a = 1 \), \( b = 14 \), \( c = 48 \).
  • Calculate the discriminant: \( \Delta = b^2 - 4ac = 4 \).
  • Substitute values into the quadratic formula: \[ c = \frac{-14 \pm \sqrt{4}}{2(1)} = \frac{-14 \pm 2}{2} \]
  • Solve for \( c \): \( c_1 = \frac{-14 + 2}{2} = -6 \) and \( c_2 = \frac{-14 - 2}{2} = -8 \).

Thus, the solutions to the quadratic equation \( c^2 + 14c + 48 = 0 \) are \( c = -6 \) and \( c = -8 \). These are the values of \( c \) that satisfy the original equation.

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Most popular questions from this chapter

Find the slope of the line that passes through \((-8,5)\) and \((-1,-3)\)

\(m^{2}+24 m+150=0\)

In the formula \(C=L+S(D+H), C\) is the number of linear inches of carpet needed to cover a staircase and landing, the number of steps on the stairs is \(S\), the depth of each step in inches is \(D\), the height of each step in inches is \(H\), and the length of the landing in inches is \(L\). A staircase has 10 steps. Each step is \(7.25\) in. deep and \(10.25\) in. high. There is one 3 -ft landing. Find the length of carpet in yards needed to cover this staircase and landing. ( \(1 \mathrm{yd}=3 \mathrm{ft} ; 1 \mathrm{ft}=12\) in.) Round up to the nearest whole number.

\(w^{2}=-3\)

\(z^{2}-20 z+164=0\)
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