Problem 30 - Algebra 2

Question

Solve each equation using the Quadratic Formula. Find the exact solutions. Then approximate any radical solutions. Round to the nearest hundredth. $$ 2 x^{2}+x=\frac{1}{2} $$

Step-by-Step Solution

Answer

The exact solutions are \(x = \frac{{-1 + \sqrt{5}}}{{4}}\) and \(x = \frac{{-1 - \sqrt{5}}}{{4}}\). The approximate solutions are -0.62 and -0.38.

Step 1: Write the equation in standard form

Get all the terms on one side of the equation so that the quadratic is set to zero: Subtract \(\frac{1}{2}\) from both sides to obtain \(2x^2 + x - \frac{1}{2} = 0\).

Step 2: Multiply all terms by 2 to eliminate the fraction

Multiplying both sides of the equation by 2 gives \(4x^2 + 2x - 1 = 0\).

Step 3: Identify the coefficients for the Quadratic Formula

In the standard quadratic form \(ax^2 + bx + c = 0\), identify the coefficients: \(a = 4\), \(b = 2\), and \(c = -1\).

Step 4: Apply the Quadratic Formula

Substitute the coefficients \(a\), \(b\), and \(c\) into the Quadratic Formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\) to find the values of \(x\).

Step 5: Calculate the discriminant

Calculate the discriminant \(\Delta = b^2 - 4ac\) which is \(\Delta = (2)^2 - 4(4)(-1) = 4 + 16 = 20\).

Step 6: Substitute the values into the Quadratic Formula

Substitute \(a = 4\), \(b = 2\), \(c = -1\), and \(\Delta = 20\) into the Quadratic Formula to find \(x\): \(x = \frac{{-2 \pm \sqrt{20}}}{{8}}\).

Step 7: Simplify the radical

Simplify \(\sqrt{20}\) by factoring it into \(\sqrt{4 \cdot 5}\) which is \(2\sqrt{5}\).

Step 8: Write the exact solutions

Write the exact solutions using the simplified radical: \(x = \frac{{-2 \pm 2\sqrt{5}}}{{8}} = \frac{{-1 \pm \sqrt{5}}}{{4}}\).

Step 9: Approximate the radical solutions

Find decimal approximations of the solutions by calculating \(x = \frac{{-1 + \sqrt{5}}}{{4}}\) and \(x = \frac{{-1 - \sqrt{5}}}{{4}}\), rounding to the nearest hundredth: First solution is approximately -0.62 and the second solution is approximately -0.38.

Key Concepts

Quadratic Formula

The Quadratic Formula is a powerful tool for solving quadratic equations, which are represented by the general form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The formula states that the solutions for x can be found with
\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]
This provides two possible solutions for x, using the plus or minus sign. The symbol ± indicates that there are typically two different solutions to a quadratic equation, one for each sign. Applying this formula involves several steps: identifying the coefficients a, b, and c, calculating the discriminant (\

Standard Form Quadratic

A quadratic equation must be in standard form to apply the Quadratic Formula. The standard form is an equation written as \[ ax^2 + bx + c = 0 \]
where a, b, and c are known values and a is not zero. This form is essential as it organizes the equation in a way that clearly displays the coefficients needed for solving the equation using the Quadratic Formula.

Converting to Standard Form

In practice, moving terms around to achieve this form can involve combining like terms, moving all terms to one side of the equality, and ensuring that a, b, and c are integer values.

Discriminant

The discriminant is a term inside the square root of the Quadratic Formula and is essential in determining the nature of the roots of a quadratic equation without actually solving it. It is represented by the equation:
\[ \Delta = b^2 - 4ac \]
The value of the discriminant helps to ascertain:

• If Δ > 0, there are two distinct real roots.
• If Δ = 0, there is one real root (also known as a repeated or double root).
• If Δ < 0, there are no real roots, but two complex roots.

The discriminant thus provides a quick way to predict the number and type of solutions for a quadratic equation.

Radical Simplification

When solving quadratic equations using the Quadratic Formula, we often encounter radical simplification. This involves simplifying the square root of numbers that often are not perfect squares. To simplify a radical:

• Factor the number inside the square root to its prime factors.
• Pair the factors if they are squares of integers.
• Simplify by taking out the square of a number as that number outside the square root.

In our example, we simplified \[ \sqrt{20} \] by writing it as \[ \sqrt{4 \cdot 5} \], which further simplifies to \[ 2\sqrt{5} \].
This process makes it easier to deal with roots and is essential when finding exact solutions for x in their simplest form.