Problem 75 - 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. $$x^{2}+0.00075 x-0.0062=0$$

Step-by-Step Solution

Answer

The solutions are approximately \(0.078\) and \(-0.079\).

Step 1 - Identify the coefficients

For the quadratic equation in standard form, which is usually written as \(ax^{2} + bx + c = 0\), identify the coefficients. Here: \[a = 1\]\[b = 0.00075\]\[c = -0.0062\]

Step 2 - Write the quadratic formula

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

Step 3 - Substitute the coefficients into the formula

Substitute \(a = 1\), \(b = 0.00075\), and \(c = -0.0062\) into the quadratic formula: \[x = \frac{{-0.00075 \pm \sqrt{{(0.00075)^2 - 4(1)(-0.0062)}}}}{2(1)}\]

Step 4 - Calculate the discriminant

First, compute the discriminant: \[(0.00075)^2 - 4(1)(-0.0062)\]\[= 0.0000005625 + 0.0248\]\[= 0.0248005625\]

Step 5 - Compute the square root of the discriminant

Now find the square root of the discriminant: \[\sqrt{0.0248005625} \approx 0.157464\]

Step 6 - Solve for the two possible values of x

Substitute the value of the square root back into the formula to get the two possible solutions: \[x_{1} = \frac{{-0.00075 + 0.157464}}{2} = \frac{0.156714}{2} = 0.078357\]\[x_{2} = \frac{{-0.00075 - 0.157464}}{2} = \frac{-0.158214}{2} = -0.079107\]Round these to three decimal places: \[x_{1} \approx 0.078\]\[x_{2} \approx -0.079\]

Step 7 - Check the solutions

Substitute \(x_{1}\approx 0.078\) and \(x_{2} \approx -0.079\) back into the original equation to verify: \[x_1: (0.078)^2 + 0.00075(0.078) - 0.0062 \approx 0\]\[x_2: (-0.079)^2 + 0.00075(-0.079) - 0.0062 \approx 0\]Both calculations should return values very close to zero, confirming the solutions are correct.

Key Concepts

solving quadratic equations

Quadratic equations have the general form \( ax^{2} + bx + c = 0 \). To solve these, we often use the quadratic formula, which is great for finding the roots, or solutions, of the quadratic equation.
Remember that in the quadratic formula: \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \] each part serves a specific purpose.
Here’s what each term means:

• a: the coefficient of \( x^{2} \).
• b: the coefficient of x.
• c: the constant term.

In our specific example, we identified these coefficients as: \[ a = 1, \ b = 0.00075, \ c = -0.0062 \].
Using the quadratic formula, we then substitute these values in place of a, b, and c. This step is crucial to correctly finding the solutions. The formula helps us handle even tricky coefficients like these.
Once we set up the equation and do the calculations, it becomes pretty straightforward. The quadratic formula makes the solution process systematic and less prone to errors.

discriminant

The discriminant is a key part of the quadratic formula found under the square root: \ b^2 - 4ac \. It helps us determine the nature of the roots of the quadratic equation.
In our example, the discriminant was calculated as: \[ (0.00075)^2 - 4 \cdot 1 \cdot (-0.0062) = 0.0000005625 + 0.0248 = 0.0248005625 \]
The value of the discriminant tells us:

• If the discriminant is positive, there are two real and distinct solutions.
• If it's zero, there’s exactly one real solution.
• If it's negative, there are no real solutions (but two complex solutions).

Here, our discriminant is positive (0.0248005625), so we have two real solutions. Calculating the square root of the discriminant ( \( \sqrt{0.0248005625} \) ) gives us an approximate value of 0.157464. This guides us in finding the specific values of x.

verifying solutions

It is important to always verify your solutions. This ensures accuracy and confirms that the computations were performed correctly.
In our example, the solutions are approximately \( x_1 \approx 0.078 \) and \( x_2 \approx -0.079 \). We can verify by substituting these values back into the original equation: \[ x^{2} + 0.00075 x - 0.0062 = 0 \] Check for \( x_1 = 0.078 \):
\( (0.078)^2 + 0.00075(0.078) - 0.0062 = 0.006084 + 0.0000585 - 0.0062 \approx 0 \)
Check for \( x_2 = -0.079 \):
\( (-0.079)^2 + 0.00075(-0.079) - 0.0062 = 0.006241 + (-0.00005925) - 0.0062 \approx 0 \)
If both values approximate to zero, our solutions are verified and correct. This final step provides assurance and confirms the accuracy of our answers.