Question

Tell how many solutions the equation has. $$ 10 x^{2}-13 x-9=0 $$

Short answer

The quadratic equation has two distinct real solutions.

Step-by-step solution

Identify the coefficients

From the equation \(10x^2 - 13x - 9 = 0\), we can identify the coefficients \(a = 10\), \(b = -13\), and \(c = -9\).

Calculate the Discriminant

Using the formula for the discriminant, \(b^2 - 4ac\), we substitute the identified coefficients: \[(-13)^2 - 4*10*(-9)\] which simplifies to \(169 + 360 = 529\).

Analyze the Discriminant

Since the discriminant \(529\) is greater than zero, the quadratic equation has two distinct real solutions.

Key concepts

Discriminant

When solving a quadratic equation, the discriminant plays a crucial role in determining the nature and number of solutions. It is represented by the formula \( b^2 - 4ac \), where \( a \) , \( b \) and \( c \) are the coefficients of the quadratic equation in the standard form \( ax^2 + bx + c = 0 \).

The value of the discriminant reveals important information about the solutions:

• If the discriminant is greater than zero, the equation has two distinct real solutions.
• If it is exactly zero, there is one real solution, also known as a repeated or double root.
• If the discriminant is less than zero, the equation has two complex solutions, which are not real numbers.

For example, in our exercise \(10x^2 - 13x - 9 = 0\), we found the discriminant to be \(529\) by squaring \(b\) and subtracting four times \(a\) and \(c\), which resulted in two distinct real solutions.

Quadratic Coefficients

The quadratic coefficients are the numerical parts that multiply each term in a quadratic equation. There are always three coefficients in the standard form \(ax^2 + bx + c = 0\):

• The first coefficient \(a\) is the coefficient of \(x^2\) and determines the equation's concavity.
• The second coefficient \(b\) is the coefficient of \(x\).
• The third coefficient \(c\) is the constant term.

Understanding these coefficients is important because they directly influence the shape and position of the parabola when the quadratic equation is graphed. Additionally, they are integral in calculations such as the discriminant or when using the quadratic formula to find solutions. In the exercise \(10x^2 - 13x - 9 = 0\), the coefficients are \(a = 10\), \(b = -13\), and \(c = -9\), which we used to calculate the discriminant.

Real Solutions

When we talk about real solutions to a quadratic equation, we refer to the x-values that satisfy the equation, which are points where the parabola crosses the x-axis. A quadratic equation can have either zero, one, or two real solutions based on the discriminant's value.

• Two distinct real solutions occur when the discriminant is positive, indicating the parabola crosses the x-axis at two different points.
• One real solution, also referred to as a double root, occurs when the discriminant is zero, meaning the parabola touches the x-axis at just one point (the vertex).
• No real solution is present when the discriminant is negative, as the parabola does not intersect the x-axis at all.

With the discriminant we computed for the exercise (\(529\)), we concluded that there are two distinct real solutions. This means our quadratic equation \(10x^2 - 13x - 9 = 0\) will intersect the x-axis at two separate points, which represent the solutions to the equation.