Problem 24 - Algebra 2 and Trigonometry

Question

In \(18-35,\) find each common solution algebraically. Express irrational roots in simplest radical form. $$ \begin{array}{l}{y=2 x^{2}-6 x+7} \\ {y=x+4}\end{array} $$

Step-by-Step Solution

Answer

The common solutions are \((3, 7)\) and \((\frac{1}{2}, \frac{9}{2})\).

Step 1: Set Equations Equal

We start by setting the two expressions for \( y \) equal to each other: \[ 2x^2 - 6x + 7 = x + 4. \] This gives us one equation to solve for \( x \).

Step 2: Simplify and Rearrange

To form a standard quadratic equation, first move all terms to one side: \[ 2x^2 - 6x + 7 - x - 4 = 0. \] Simplify the equation further: \[ 2x^2 - 7x + 3 = 0. \]

Step 3: Use Quadratic Formula

The quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) can be used to solve the equation. Here, \( a = 2 \), \( b = -7 \), and \( c = 3 \). Substitute these into the formula: \[ x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4 \times 2 \times 3}}{2 \times 2}. \]

Step 4: Calculate Discriminant

Evaluate the discriminant inside the square root: \[ b^2 - 4ac = (-7)^2 - 4 \cdot 2 \cdot 3 = 49 - 24 = 25. \]

Step 5: Solve for x

Substitute the discriminant into the quadratic formula: \[ x = \frac{7 \pm \sqrt{25}}{4}. \] Since \( \sqrt{25} = 5 \), the solutions are: \[ x = \frac{7 + 5}{4} \quad \text{or} \quad x = \frac{7 - 5}{4}. \] This simplifies to \( x = 3 \) or \( x = \frac{1}{2} \).

Step 6: Find Corresponding y-values

Substitute \( x = 3 \) and \( x = \frac{1}{2} \) back into one of the original equations, say \( y = x+4 \): For \( x = 3 \), \[ y = 3 + 4 = 7. \] For \( x = \frac{1}{2} \), \[ y = \frac{1}{2} + 4 = \frac{9}{2}. \]

Key Concepts

Irrational Roots

Quadratic equations can sometimes have irrational roots, meaning the solutions to the equation are not whole numbers or fractions, but instead involve square roots of non-perfect squares which cannot be resolved into simpler numeric forms. An irrational root often appears in quadratic equations when the discriminant (inside the square root of the quadratic formula) is not a perfect square.

For instance, if the discriminant is a perfect square like 16 or 25, the roots would be rational. However, if it were something like 7 or 13, the roots would involve irrational numbers as you can't simplify the square root of 7 or 13 into a neat number.
When dealing with irrational roots in equations, it’s common to leave them in what's known as the simplest radical form for clarity and precision.

Quadratic Formula

The Quadratic Formula is a powerful tool used to find roots of a quadratic equation of the form \( ax^2 + bx + c = 0 \). The formula is given as:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\]
This formula directly utilizes the coefficients of the quadratic equation:

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

It calculates both potential solutions by accounting for the \(\pm\) symbol, which implies that two different values for \(x\) could satisfy the equation. Each solution corresponds to either the addition or subtraction of the square root in the formula.

This formula works universally for all quadratic equations, providing solutions even when roots are complex or irrational.

Discriminant

The discriminant is a fundamental part of the quadratic formula, crucial in determining the nature of the roots. Represented by \(D = b^2 - 4ac\), the value of the discriminant tells us:

• If \(D > 0\), there are two distinct real roots.
• If \(D = 0\), there is exactly one real root, known as a repeated or double root.
• If \(D < 0\), there are no real roots; instead, the roots are complex or imaginary.

In the given exercise, the discriminant calculated was 25, which is a positive perfect square, indicating two distinct rational roots. This makes the solution process smoother, without involving irrational numbers, as the square root of 25 is straightforward.

Understanding and calculating the discriminant efficiently tells you the kind of solutions to expect, adjusting your solving strategy accordingly.

Simplest Radical Form

Simplest radical form refers to expressing a number or solution containing a square root in its most reduced state. This ensures that any square roots are simplified as much as possible. For instance, the square root of 50 simplifies to \(5\sqrt{2}\) because 50 can be broken down into 25 times 2, and the square root of 25 is 5.

Keeping expressions in simplest radical form is valuable when solutions involve irrational roots. It maintains accuracy without resorting to decimal approximations, which can be less precise.
In our exercise, the discriminant was a perfect square, so it resulted in rational roots. If it were not a perfect square, leaving the result in simplest radical form ensures the solution remains exact.

Practice simplifying radicals by breaking numbers down into their prime factors and pulling out squares. This skill is essential when dealing with quadratic equations with non-perfect squares.