Problem 71 - Algebra for College Students

Question

Use the quadratic formula and a calculator to solve each equation. Round answers to three decimal places and check your answers. $$1.85 x^{2}+6.72 x+3.6=0$$

Step-by-Step Solution

Answer

The solutions are \(x \approx -0.653\) and \(x \approx -2.979\).

Step 1: Identify coefficients

For the quadratic equation in the form \[ax^2 + bx + c = 0\] identify the coefficients. Here, \[a = 1.85, \ b = 6.72, \ c = 3.6\]

Step 2: Write down the quadratic formula

The quadratic formula is given by \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Step 3: Calculate the discriminant

Compute the discriminant using the formula \[b^2 - 4ac\] Here, \[b^2 = 6.72^2 = 45.1584, \ 4ac = 4 \cdot 1.85 \cdot 3.6 = 26.64\]Thus, the discriminant is \[45.1584 - 26.64 = 18.5184\]

Step 4: Compute the square root of the discriminant

Calculate \(\sqrt{18.5184} = 4.304\)

Step 5: Solve for x using the positive square root

Substitute the positive value of the square root of the discriminant and the coefficients into the quadratic formula: \[x = \frac{-6.72 + 4.304}{2 \cdot 1.85} = \frac{-2.416}{3.7} = -0.653\]

Step 6: Solve for x using the negative square root

Substitute the negative value of the square root of the discriminant and the coefficients into the quadratic formula: \[x = \frac{-6.72 - 4.304}{2 \cdot 1.85} = \frac{-11.024}{3.7} = -2.979\]

Step 7: Check the solutions

Substitute the solutions back into the original equation to verify: For \(x = -0.653\)\(1.85(-0.653)^2 + 6.72(-0.653) + 3.6 = 0\) This simplifies to approximately 0.For \(x = -2.979\)\(1.85(-2.979)^2 + 6.72(-2.979) + 3.6 = 0\) This also simplifies to approximately 0.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations, which are equations in the form \(ax^2 + bx + c = 0\). It provides a straightforward way to find the roots (solutions) regardless of whether the roots are real or complex. The formula is expressed as: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This equation might look intimidating, but it's simply a matter of plugging in the coefficients \(a\), \(b\), and \(c\) from your quadratic equation.

Discriminant

The discriminant is a part of the quadratic formula that helps determine the nature of the roots. It's given by \(b^2 - 4ac\). The value of the discriminant can tell us if the roots of the quadratic equation are real or complex:

• If the discriminant \(b^2 - 4ac\) is positive, the equation has two distinct real roots.
• If it is zero, there is exactly one real root (also called a repeated root).
• If the discriminant is negative, the equation has two complex roots.

In the given exercise, the discriminant is \(18.5184\), which is positive, indicating that there are two distinct real roots.

Verifying Solutions

After calculating the solutions using the quadratic formula, it's crucial to verify them by substituting the values back into the original equation to ensure they satisfy it. This step confirms the accuracy of your solutions. In this exercise, we find the roots to be \(-0.653\) and \(-2.979\).
When we substitute \(-0.653\) into the equation \(1.85(-0.653)^2 + 6.72(-0.653) + 3.6\), it indeed simplifies to approximately 0. Similarly, substituting \(-2.979\) confirms that it is also a solution.

Rounding Decimals

Rounding decimals to a certain number of places can make numbers easier to work with, especially in practical applications. In this exercise, we round the solutions to three decimal places. This means you keep three digits after the decimal point. If you're unsure how to round, follow these steps:

• Identify the digit at the place you need to round to (third place here).
• Look at the next digit to the right.
• If it's 5 or more, round the identified digit up by one.
• If it's 4 or less, leave the identified digit as it is.

For example, for the number \(4.304\), after rounding to three decimal places, it remains \(4.304\) because the fourth decimal is already zero.