Question

Find the real solutions, if any, of each equation. $$ 2 x^{2}-13 x+21=0 $$

Short answer

The solutions are x = 3 and x = 3.5.

Step-by-step solution

Identify Coefficients

The given quadratic equation is \[2x^2 - 13x + 21 = 0\]. Identify the coefficients: \[a = 2\], \[b = -13\], and \[c = 21\].

Apply the Quadratic Formula

Use the quadratic formula \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. Plug in the coefficients: \[x = \frac{-(-13) \pm \sqrt{(-13)^2 - 4 \cdot 2 \cdot 21}}{2 \cdot 2}\].

Simplify Inside the Square Root

Calculate inside the square root: \[(-13)^2 - 4 \cdot 2 \cdot 21 = 169 - 168 = 1\].

Evaluate the Square Root

Evaluate the square root: \[\sqrt{1} = 1\].

Calculate the Solutions

Substitute back into the formula: \[x = \frac{13 \pm 1}{4}\]. Calculate the two possible values of x: \[x = \frac{13 + 1}{4} = \frac{14}{4} = 3.5\] and \[x = \frac{13 - 1}{4} = \frac{12}{4} = 3\].

Write Down the Solutions

The real solutions to the equation are \[x = 3\] and \[x = 3.5\].

Key concepts

solving quadratic equations

To solve quadratic equations, you need to find the values of the variable that make the equation true. Quadratic equations are expressions of the form \[ax^2 + bx + c = 0\] where \(a\), \(b\), and \(c\) are constants. Identifying these coefficients is critical in applying any solving technique. The quadratic formula, factoring, and completing the square are popular methods to solve these equations. In this example, we use the quadratic formula, a powerful tool that works for any quadratic equation.To get started:

• Identify the coefficients \(a\), \(b\), and \(c\) from the given equation.
• Substitute these values into a specific formula to find the solutions.

Let's go deeper into using the quadratic formula for solving quadratic equations.

quadratic formula

The quadratic formula is a universal method for finding the roots of any quadratic equation. The formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, the terms under the square root symbol and their calculation are crucial. The term \(b^2 - 4ac\) is called the discriminant and determines the nature of the solutions.

Follow these steps when using the quadratic formula:

• Substitute the identified values of \(a\), \(b\), and \(c\) into the formula.
• Simplify inside the square root (discriminant).
• Evaluate the square root.
• Calculate the solutions using both the plus and minus operations.

Let's apply this to our example:

1. Substitute \(a = 2\), \(b = -13\), and \(c = 21\) into the formula:
2. \(x = \frac{-(-13) \pm \sqrt{(-13)^2 - 4 \cdot 2 \cdot 21}}{2 \cdot 2}\)
3. Calculate the discriminant: \((-13)^2 - 4 \cdot 2 \cdot 21 = 169 - 168 = 1\)
4. Find the square root of 1, which is 1.
5. Complete the formula to find the solutions:\[x = \frac{13 \pm 1}{4}\]
6. This gives \(x = 3\) and \(x = 3.5\).

The solutions are \(x = 3\) and \(x = 3.5\).

real solutions

Real solutions are the values of \(x\) that solve the quadratic equation and are real numbers. The discriminant \(b^2 - 4ac\) helps determine the nature of the solutions.

• If the discriminant is positive, there are two distinct real solutions.
• If the discriminant is zero, there is one real solution (a repeated root).
• If the discriminant is negative, there are no real solutions (the solutions are complex numbers).

In our example, the discriminant is 1, which is positive. Therefore, we have two distinct real solutions: \(x = 3\) and \(x = 3.5\).These solutions are points where the quadratic equation \(2x^2 - 13x + 21 = 0\) crosses the x-axis. Evaluating real solutions is essential in understanding how quadratic equations behave on a graph.