Problem 36 - Elementary Algebra

Question

Use the quadratic formula to solve. $$ x 2+10 x+23=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = -5 + \sqrt{2}\) and \(x = -5 - \sqrt{2}\).

Step 1: Identify the coefficients

The given quadratic equation is \(x^2 + 10x + 23 = 0\). Identify the coefficients: \(a = 1\), \(b = 10\), and \(c = 23\).

Step 2: Write the quadratic formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula will be used to find the roots of the quadratic equation.

Step 3: Calculate the discriminant

Compute the discriminant \(b^2 - 4ac\). Substituting the values, we get \(10^2 - 4 \times 1 \times 23 = 100 - 92 = 8\).

Step 4: Solve using the quadratic formula

Substitute the coefficients and the discriminant into the quadratic formula: \[x = \frac{-10 \pm \sqrt{8}}{2 \times 1}\] Simplify the expression inside the square root, \( \sqrt{8} = 2\sqrt{2}\). This gives us \[x = \frac{-10 \pm 2\sqrt{2}}{2}\].

Step 5: Simplify the expression

Split the expression to simplify: \[x = \frac{-10}{2} \pm \frac{2\sqrt{2}}{2}\] This simplifies to \[x = -5 \pm \sqrt{2}\].

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation with the highest degree of 2. It generally takes the form \(ax^2 + bx + c = 0\), where:

• \(a\), \(b\), and \(c\) are constants, with \(a eq 0\)
• \(x\) represents the variable or unknown we're solving for

Quadratic equations are very common in algebra and have various methods for solving them, such as factoring, completing the square, and using the quadratic formula.
The quadratic formula offers a systematic approach to finding the roots of any quadratic equation, especially useful when the equation is not easily factorable.
In our problem, \(x^2 + 10x + 23 = 0\), we notice that it's already in standard form. We just need to identify \(a = 1\), \(b = 10\), and \(c = 23\) to proceed with solving it.

Discriminant

The discriminant is part of the quadratic formula found under the square root symbol: \(b^2 - 4ac\). It's crucial because it tells you about the nature of the roots without having to solve them completely. Here's what it indicates:

• If \(b^2 - 4ac > 0\), the equation has two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root (a repeated root).
• If \(b^2 - 4ac < 0\), the roots are complex and not real.

In our example, we compute the discriminant of \(x^2 + 10x + 23 = 0\) as \(8\), indicating the equation has two distinct real roots. Knowing this helps us predict the nature of solutions without fully solving the equation yet.

Roots of the Equation

The roots of a quadratic equation are the solutions that satisfy the equation, effectively making it equal to zero. They can be found using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] This formula cleverly combines all the identified parameters of the equation into a single solution step.
In the equation \(x^2 + 10x + 23 = 0\), we substitute \(a = 1\), \(b = 10\), and \(c = 23\) along with the discriminant we calculated: \[x = \frac{-10 \pm \sqrt{8}}{2}\] On simplification, this step breaks down to get the actual roots: \[x = -5 \pm \sqrt{2}\] This outcome gives us two roots: \(x = -5 + \sqrt{2}\) and \(x = -5 - \sqrt{2}\). Hence, these roots result from solving the quadratic equation using its respective formula and show how versatile and useful the formula can be for different types of quadratic problems.