Problem 51 - Algebra 1: Concepts and Skills

Question

Solve the quadratic equation. $$ x^{2}+20 x+2=0 $$

Step-by-Step Solution

Answer

The solutions to the equation are \( x1 = -10 - \sqrt{98} \) and \( x2 = -10 + \sqrt{98} \).

Step 1 Identify the parameters

Firstly, identify the coefficients in the equation \( ax^2 + bx + c = 0 \). In our case, \( a = 1 \), \( b = 20 \), and \( c = 2 \).

Step 2 Calculate the Discriminant

The discriminant is found using the formula \( D = b^2 - 4ac \). Substitute \( b = 20 \), \( a = 1 \), and \( c = 2 \) into the formula to get \( D = (20)^2 - 4*1*2 = 400 - 8 = 392 \).

Step 3 Apply Quadratic Formula

Now, apply the quadratic formula, which is \( x = (-b \pm \sqrt{D}) / (2a) \). This gives us the two roots \( x1 = (-20 - \sqrt{392}) / 2 = -10 - \sqrt{98} \), and \( x2 = (-20 + \sqrt{392}) / 2 = -10 + \sqrt{98} \).

Key Concepts

Understanding the Discriminant

The discriminant is a key concept when it comes to solving quadratic equations. It helps us understand what type of solutions we can expect from the equation. The discriminant is calculated using the formula:

• \( D = b^2 - 4ac \)

In a quadratic equation in the form \( ax^2 + bx + c = 0 \), \( a \), \( b \), and \( c \) are the coefficients. Once these values are known, you plug them into the formula to find the discriminant.

Here's what the discriminant tells us:

• If \( D > 0 \), there are two real and distinct roots.
• If \( D = 0 \), there is one real root, or the roots are repeated.
• If \( D < 0 \), there are no real roots, but two complex roots.

For our equation \( x^2 + 20x + 2 = 0 \), the discriminant \( D \) is 392, which is greater than zero. This tells us we will have two real and distinct roots.

The Quadratic Formula

The quadratic formula is a powerful tool that provides the solution to any quadratic equation. Once you've determined your discriminant, \( D \), you can proceed to find the roots of the equation using this formula:

• \( x = \frac{-b \pm \sqrt{D}}{2a} \)

This formula helps in straightforwardly finding the values of \( x \) that make the equation true. Simply substitute \( a \), \( b \), and \( D \) into the formula.

It includes the following components:

• \(-b\) is the negative of the coefficient of \( x \).
• \(\sqrt{D}\) is the square root of the discriminant.
• \(2a\) is twice the coefficient of \( x^2 \).

By applying the quadratic formula to our equation, you calculate the roots which are \( x_1 = \frac{-20 - \sqrt{392}}{2} \) and \( x_2 = \frac{-20 + \sqrt{392}}{2} \).

Roots of Quadratic Equations

Finding the roots of a quadratic equation involves solving for values of \( x \) that satisfy the equation. Once we apply the quadratic formula, we obtain the roots or the solutions of the equation, which are essentially the x-values where the quadratic graph intersects the x-axis.

For the equation \( x^2 + 20x + 2 = 0 \), once the discriminant and the quadratic formula are applied, the roots are alongside complex numbers giving two real numbers:

• \( x_1 = -10 - \sqrt{98} \)
• \( x_2 = -10 + \sqrt{98} \)

These roots depict the x-intercepts of the parabola described by our quadratic equation. Each root represents a unique solution to the equation, highlighting points where the curve of the quadratic equation meets the x-axis.

Understanding these roots helps in visualizing the behavior of the quadratic graph.