Problem 8 - A Graphical Approach to College Algebra

Question

Solve each problem. Do not use a calculator. $$\text { Solve } \frac{1}{4} x^{2}+x=1$$

Step-by-Step Solution

Answer

The solutions are \(x = -2 + 2\sqrt{2}\) and \(x = -2 - 2\sqrt{2}\).

Step 1: Move all terms to one side

We begin by rearranging the equation so that all terms are on one side, set to zero. This will form a quadratic equation. Start by subtracting 1 from both sides: \[\frac{1}{4}x^2 + x - 1 = 0\]

Step 2: Clear the fraction

Multiply every term by 4 to eliminate the fraction. This gives:\[x^2 + 4x - 4 = 0\]

Step 3: Identify the quadratic coefficients

The quadratic equation is now in the standard form \(ax^2 + bx + c = 0\). Identify the coefficients: - \(a = 1\)- \(b = 4\)- \(c = -4\).

Step 4: Use the quadratic formula

The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) is used to find the roots (solutions) of the equation. Substitute the values:\[x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-4)}}{2(1)}\]

Step 5: Simplify the expression under the square root

Calculate the discriminant (the expression under the square root):- \(b^2 = 16\)- \(4ac = 4 \times 1 \times (-4) = -16\)- \(b^2 - 4ac = 16 + 16 = 32\).So, the formula is:\[x = \frac{-4 \pm \sqrt{32}}{2}\]

Step 6: Simplify the square root

Notice that \(\sqrt{32}\) can be simplified:- \(\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}\).So, the formula becomes:\[x = \frac{-4 \pm 4\sqrt{2}}{2}\]

Step 7: Simplify the entire expression

Divide each term by 2:- \(x = \frac{-4}{2} \pm \frac{4\sqrt{2}}{2}\)- \(x = -2 \pm 2\sqrt{2}\).Thus, the solutions are \(x = -2 + 2\sqrt{2}\) and \(x = -2 - 2\sqrt{2}\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation. A quadratic equation is of the form \(ax^2 + bx + c = 0\). Using the quadratic formula allows you to find the values of \(x\) that satisfy this equation. The formula is:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula works because it derives its logic from the process of completing the square. To use the quadratic formula, follow these steps:

• Identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation.
• Substitute these values into the formula.
• Simplify the resulting expression to find the solutions for \(x\).

The formula includes a square root, indicating that quadratic equations can have two solutions, indicated by the \(\pm\) symbol. This means you will have to consider both the addition and subtraction of the term \(\sqrt{b^2 - 4ac}\). This inclusion allows for finding two possible values of \(x\) that make the statement true.

Discriminant

The discriminant is part of the quadratic formula and plays a crucial role in determining the nature of the solutions. It is the expression under the square root: \(b^2 - 4ac\). Understanding the value of the discriminant can tell you:

• How many real solutions exist.
• The nature of the solutions.

If \(b^2 - 4ac > 0\), the equation has two distinct real roots. If \(b^2 - 4ac = 0\), there is exactly one real root, meaning the parabola touches the x-axis at one point. If \(b^2 - 4ac < 0\), there are no real roots, but two complex roots. In our exercise, the discriminant was found to be 32. Since 32 is positive, it tells us that there are two distinct real solutions for the equation. This insight helps not only in solving but also in understanding the graphical representation of quadratic equations.

Simplifying Expressions

Simplifying expressions is a fundamental skill, especially in solving quadratic equations. It involves breaking down more complex expressions into simpler terms to make calculations easier. When using the quadratic formula, it is essential to simplify both the square root and the resulting fractions whenever possible.During the exercise, the expression under the square root becomes \(\sqrt{32}\). \(\sqrt{32}\) can be simplified by recognizing it as a product of squares: \(\sqrt{16 \times 2} = 4\sqrt{2}\). This simplification reduces calculation complexity, leading to clearer and more manageable steps.Furthermore, the solution expressions as \(-2 \pm 2\sqrt{2}\) were simplified after dividing each term by 2. Simplification ensures the solution is presented in its most straightforward form, which is not only neat but also more interpretable. Each simplification step serves to cut complexity and provide a transparent view of the solutions, a vital part of proficiently working with algebraic expressions.