Problem 11 - Algebra 1: Concepts and Skills

Question

Use the quadratic formula to solve the equation. Write your solutions in simplest form. $$3 x^{2}+x-1=0$$

Step-by-Step Solution

Answer

The solutions to the equation in simplest form are \(x = \frac{-1 + \sqrt{13}}{6}\) and \(x = \frac{-1 - \sqrt{13}}{6}\)

Step 1: Identify the coefficients

The given quadratic equation is \(3 x^{2}+x-1=0\). In this equation, the coefficients are \(a=3\), \(b=1\), and \(c=-1\).

Step 2: Use the quadratic formula

Substitute the coefficients into the quadratic formula. \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] This simplifies to: \[x = \frac{-1 \pm \sqrt{1^{2} - 4(3)(-1)}}{2(3)}\]

Step 3: Simplify the expression

Simplify the equation under the square root and the denominator: \[x = \frac{-1 \pm \sqrt{1 +12}}{6}\] Simplify further to get: \[x = \frac{-1 \pm \sqrt{13}}{6}\] which are the solutions in simplest form.

Key Concepts

Solving Quadratic Equations

Quadratic equations are a core component of algebra, commonly written in the form \(ax^2 + bx + c = 0\). Solving these equations requires finding the values of \(x\) that make the equation true. One powerful method for solving quadratic equations is the quadratic formula. This method can solve any quadratic equation, regardless of whether it can be factored easily.

The quadratic formula is expressed as:

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

In this formula, \(-b\) represents the opposite of the coefficient \(b\), while \(\pm\) indicates that there are generally two possible solutions, due to the nature of squaring in quadratic equations.

By using the quadratic formula, you substitute the values of the coefficients (\(a\), \(b\), and \(c\)) directly into the formula. Then, perform arithmetic operations as indicated to solve for \(x\). This method is particularly useful when the quadratic equation cannot be easily simplified or factored.

Quadratic Equation Coefficients

In a quadratic equation of the form \(ax^2 + bx + c = 0\), the letters \(a\), \(b\), and \(c\) represent coefficients. These coefficients are crucial because they define the shape of the parabola represented by the quadratic equation.

• \(a\) is the coefficient of \(x^2\) and determines the width and direction of the parabola. If \(a\) is positive, the parabola opens upwards; if negative, it opens downwards.
• \(b\) is the coefficient of \(x\), which affects the position of the vertex of the parabola along the x-axis.
• \(c\) is the constant term and sets the y-intercept, the point where the parabola crosses the y-axis.

In our example equation \(3x^2 + x - 1 = 0\), the coefficients are \(a=3\), \(b=1\), and \(c=-1\). To solve this quadratic equation, these coefficients are substituted into the quadratic formula. Understanding the role of each coefficient helps predict the behavior of the function and effectively solve the equation.

Simplifying Square Roots

When solving quadratic equations using the quadratic formula, you often encounter square roots. Simplifying square roots helps get the final solution in its simplest form. Simplification makes calculations easier and solutions more comprehensible.

Consider the equation \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. Inside the square root, the expression \(b^2 - 4ac\) is called the discriminant. The value of the discriminant can tell us about the nature of the roots:

• If the discriminant is positive, we have two real and distinct solutions.
• If it is zero, there is one real solution.
• If negative, the solutions are complex numbers.

In our solution, \(b^2 - 4ac\) results in \(1 - 12 = -11\), which simplifies to \(\sqrt{13}\) in our computed solution. Simplifying square roots accurately is key to finding the correct values of \(x\) and ensuring the solutions remain exact and clear.