Problem 38 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation algebraically. Check your solution graphically. $$10 x^{2}-23 x-5=0$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 2.5\) and \(x = -0.2\).

Step 1: Applying the Quadratic Formula

To solve this quadratic equation we apply the quadratic formula, which is \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\), where \(a\), \(b\), and \(c\) are the coefficients from the equation \(ax^{2}+bx+c=0\). For our equation, \(10x^{2}-23x-5=0\), \(a=10\), \(b=-23\) and \(c=-5\)

Step 2: Calculating the Discriminant

First, let's calculate the discriminant, which is \(b^{2}-4ac\). Substituting \(a=10\), \(b=-23\) and \(c=-5\) gives a discriminant of \((-23)^{2}-4*10*(-5) = 729\)

Step 3: Finding the Solutions

Substitute \(a=10\), \(b=-23\) and the discriminant into the quadratic formula: \(x = \frac{-(-23) \pm \sqrt{729}}{20}\). Simplifying this gives \(x = \frac{23 \pm 27}{20}\), so the solutions are \(x = 2.5\) and \(x = -0.2\)

Step 4: Checking the Solution Graphically

To check solution graphically, plot the function \(f(x) = 10x^{2}-23x-5\). You should find that the graph of the function intersects the x-axis at \(x = 2.5\) and \(x = -0.2\), confirming the solutions.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations of the form \(ax^{2} + bx + c = 0\). It provides the solutions, or roots, of the quadratic equation using the coefficients \(a\), \(b\), and \(c\).
By substituting these coefficients into the formula, \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\), we get the potential solutions for \(x\). The use of the \(\pm\) sign indicates that there can be two solutions or roots, depending on the discriminant. For the equation \(10x^{2} - 23x - 5 = 0\), we identify \(a = 10\), \(b = -23\), and \(c = -5\), and insert these values into the formula to solve for \(x\).
The quadratic formula is useful when a quadratic equation cannot be factored easily. It provides a straightforward algebraic solution, ensuring that no possible solutions are overlooked.

Discriminant

The discriminant is a key part of the quadratic formula and gives information about the nature of the roots of the quadratic equation. It is represented by \(b^{2} - 4ac\).
The value of the discriminant determines the number and type of solutions to the quadratic equation:

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root, also known as a repeated or double root.
• If it is negative, there are no real roots, but two complex roots.

In the given problem, substituting \(a = 10\), \(b = -23\), and \(c = -5\) into the discriminant gives \((-23)^{2} - 4 \times 10 \times (-5) = 729\).
Since the discriminant is positive, the quadratic equation has two distinct real solutions, confirming its graphical solution.

Graphical Solution

Graphing a quadratic equation offers a visual way to confirm solutions found algebraically. The graph of a quadratic equation is a parabola, which can either open upwards or downwards depending on the sign of \(a\).
The solutions, or roots, of the equation are the x-values where the parabola intersects the x-axis. For the equation \(f(x) = 10x^{2} - 23x - 5\), graphing the function provides a visual check. It should intersect the x-axis at the solutions \(x = 2.5\) and \(x = -0.2\).
If the parabola does not intersect the x-axis, this indicates no real solutions exist for the equation. Graphical solutions are excellent for validating algebraic results and for understanding the behavior of the quadratic function visually.

Algebraic Solution

An algebraic solution to a quadratic equation involves manipulating the equation to isolate the variable \(x\), most often through methods such as factoring, completing the square, or using the quadratic formula.
In the case of a more complex quadratic equation like \(10x^{2} - 23x - 5 = 0\), the quadratic formula is often the most efficient method.

• Start by identifying the coefficients: \(a = 10\), \(b = -23\), \(c = -5\).
• Calculate the discriminant: \( (-23)^{2} - 4 \times 10 \times (-5) = 729\).
• Substitute into the formula: \(x = \frac{23 \pm \sqrt{729}}{20}\), simplifying to find the roots \(x = 2.5\) and \(x = -0.2\).

Using algebraic methods helps ensure a comprehensive understanding of the solution process. It also allows us to solve for precise values even if the equation isn't easily factorable.