Problem 84 - Algebra 1: Concepts and Skills

Question

Tell whether the equation has two solutions, one solution, or no real solution. $$ 10 x^{2}-13 x-9=0 $$

Step-by-Step Solution

Answer

The given equation has two distinct real solutions.

Step 1: Identifying the coefficients

Identify the values for a, b, and c in the quadratic equation. Here, \(a = 10\), \(b = -13\), and \(c = -9\).

Step 2: Calculate the Discriminant

The Discriminant is given by \(D = b^2 - 4ac\). Substitute the values of a, b, and c and calculate the value. Thus, \(D = (-13)^2 - 4*10*(-9) = 169 + 360 = 529\).

Step 3: Interpret the result

Since the Discriminant is greater than 0 (D=529), therefore, the given equation has two distinct real solutions.

Key Concepts

Discriminant in Quadratic Equations

Understanding the discriminant is crucial when solving quadratic equations. In simple terms, the discriminant is a part of the quadratic formula and it tells us about the nature of the roots without actually solving the equation. The formula to find the discriminant (D) is \(D = b^2 - 4ac\), where \(a\), \(b\), and \(c\) are the coefficients of the terms \(ax^2\), \(bx\), and \(c\) respectively in the general form of a quadratic equation \(ax^2 + bx + c = 0\).

Here's how the discriminant informs us about the solutions:

• If \(D > 0\), the quadratic equation has two distinct real roots.
• If \(D = 0\), there is exactly one real root, meaning that the quadratic has a single solution where both roots are the same.
• If \(D < 0\), the equation has no real roots; instead, it has two complex roots.

So, by simply calculating the value of the discriminant, we save time and effort for students to determine the number and type of solutions a quadratic equation has.

Finding Coefficients in Quadratics

When solving quadratic equations, identifying the coefficients correctly is the first step. Coefficients are the numerical parts of the terms of a polynomial. A standard form of a quadratic equation is \(ax^2 + bx + c = 0\). Here each letter represents a coefficient: \(a\) is the coefficient of \(x^2\), \(b\) is the coefficient of \(x\), and \(c\) is the constant term. It's essential for students to be able to find these coefficients because they are used in various methods for solving the quadratic equation, such as factoring, completing the square, or using the quadratic formula.

An easy way to remember this is to think of \(a\), \(b\), and \(c\) as placeholders in the equation that hold the specific numerical values which will affect the shape and position of the parabola represented by the quadratic equation. Identifying these coefficients correctly will lead to a successful solution of the quadratic.

Solving Quadratic Equations

There are multiple ways to solve quadratic equations, and one's choice may depend on the specific form of the quadratic equation involved. The three primary methods are:

• Factoring: If the quadratic factors nicely, it can be broken down into two binomials which when set equal to zero, give the roots of the equation.
• Completing the Square: This method involves rearranging the equation to form a perfect square trinomial, making it easy to solve for \(x\).
• Quadratic Formula: This method uses the coefficients \(a\), \(b\), and \(c\) to find the equation's roots by plugging them into the formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\).

Remember, for the quadratic formula to work, the equation must be in the standard form \(ax^2 + bx + c = 0\). By first identifying the coefficients and then calculating the discriminant, students can choose the most efficient method to find the roots to save time and effort.