Problem 53
Question
\(c^{2}+14 c+48=0\)
Step-by-Step Solution
Verified Answer
The solutions are \( c = -6 \) and \( c = -8 \).
1Step 1 - 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 \(c = 48\).
2Step 2 - 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.
3Step 3 - Calculate the discriminant
Calculate the discriminant (\( \Delta \)) using:\[ \Delta = b^2 - 4ac \]Substitute the values: \[ \Delta = 14^2 - 4(1)(48) = 196 - 192 = 4 \]
4Step 4 - 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:\[ c_1 = \frac{-14 + 2}{2} = -6 \]\[ c_2 = \frac{-14 - 2}{2} = -8 \]
5Step 5 - Write the final answer
The solutions to the quadratic equation \( c^2 + 14c + 48 = 0 \) are \( c = -6 \) and \( c = -8 \).
Key Concepts
quadratic formuladiscriminantsolving quadratic equations
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:
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.
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
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:
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.
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:
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.
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.