Problem 53 - Algebra 1

Question

Solve the quadratic equation. $$x^{2}-4 x-6=0$$

Step-by-Step Solution

Answer

The solutions to the equation \( x^{2}-4x-6=0 \) are \( x=2+\sqrt{10} \) and \( x=2-\sqrt{10} \).

Step 1: Identify values of a, b and c

In the given equation \( x^{2}-4x-6=0 \), \( a=1 \), \( b=-4 \), and \( c=-6 \) are the coefficients of the quadratic equation.

Step 2: Substitute values into the quadratic formula

Now, substitute the values of \( a \), \( b \), and \( c \) into the quadratic formula \( x=\frac{-b \pm \sqrt{b^{2}-4ac}}{2a} \). So \( x=\frac{-(-4) \pm \sqrt{(-4)^{2}-4*1*(-6)}}{2*1} \).

Step 3: Simplifying the expression

Simplifying the expression gives the solution \( x=\frac{4 \pm \sqrt{16+24}}{2} \). Which further simplifies to \( x=\frac{4 \pm \sqrt{40}}{2} \). After simplification, we get \( x=\frac{4 \pm 2\sqrt{10}}{2} \). This simplifies further to \( x= 2 \pm \sqrt{10} \). Thus, the roots of the given quadratic equation are \( x=2+\sqrt{10} \) and \( x=2-\sqrt{10} \).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of quadratic equations. A quadratic equation typically takes the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The quadratic formula itself is given by:

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

This formula allows us to calculate the exact values of \( x \) that satisfy the equation, often referred to as the roots or solutions of the equation.
By plugging the values of \( a \), \( b \), and \( c \) from the equation into this formula, we directly get to the roots. In this specific equation, \( a = 1 \), \( b = -4 \), and \( c = -6 \).
After substitution, most of the work involves simplifying the expression under the square root, known as the discriminant \( b^2 - 4ac \), which determines the nature of the roots: real and distinct, real and equal, or complex.

Roots of Equations

The roots of a quadratic equation are the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). Visualize these roots as the x-intercepts where the graph of the quadratic function crosses the x-axis.
For our specific example, we found the roots by setting the quadratic expression \( x^2 - 4x - 6 = 0 \) equal to zero and solving for \( x \) using the quadratic formula. Our calculated roots were \( x = 2 + \sqrt{10} \) and \( x = 2 - \sqrt{10} \).

• The expression \( \sqrt{10} \) illustrates that the roots are irrational, meaning they cannot be expressed as simple fractions.
• Such roots will produce a parabola on the graph that does not touch integer x-values at its roots.

Understanding why quadratic equations have two roots, and how to calculate them, are key skills when dealing with polynomials.

Simplifying Expressions

Simplifying expressions is an essential step in solving quadratic equations. It involves breaking down the expression into a more manageable or easily understandable form. For the equation \( x^2 - 4x - 6 = 0 \), we used the quadratic formula and aimed to simplify the expression for \( x \).

• We started by calculating \( b^2 - 4ac \), which became \( 16 + 24 = 40 \).
• Next, calculating \( \sqrt{40} \) gave us \( 2\sqrt{10} \), as we factor out the perfect square 4 from 40.
• Then, the full expression \( \frac{4 \pm 2\sqrt{10}}{2} \) was simplified step-by-step by dividing both terms in the numerator by 2.

This careful simplification led to the final outcome, \( x = 2 \pm \sqrt{10} \), providing the solution in its simplest and most interpretable form for further analysis or graphing.