Problem 14 - Beginning and Intermediate Algebra

Question

Use the quadratic formula to solve each equation. (All solutions for these equations are real numbers.) $$ 2 x^{2}+3 x-1=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{-3 + \sqrt{17}}{4}\) and \(x = \frac{-3 - \sqrt{17}}{4}\).

Step 1 - Identify the coefficients

For the quadratic equation in the form of \(ax^2 + bx + c = 0\), identify the coefficients \(a\), \(b\), and \(c\). In this case, \(a = 2\), \(b = 3\), and \(c = -1\).

Step 2 - Write the quadratic formula

Recall the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. Substitute the identified coefficients \(a\), \(b\), and \(c\) into the quadratic formula.

Step 3 - Calculate the discriminant

Compute the discriminant using the formula \(b^2 - 4ac\). For our coefficients, \(3^2 - 4(2)(-1) = 9 + 8 = 17\).

Step 4 - Substitute into the quadratic formula

Substitute the coefficients and the discriminant value back into the quadratic formula: \[x = \frac{-3 \pm \sqrt{17}}{4}\].

Step 5 - Compute the solutions

Use the quadratic formula to find the two solutions: \[x = \frac{-3 + \sqrt{17}}{4}\] and \[x = \frac{-3 - \sqrt{17}}{4}\]. These are the solutions to the quadratic equation.

Key Concepts

solving quadratic equations

Quadratic equations are polynomials of the form \[ax^2 + bx + c = 0\], where \(a\), \(b\), and \(c\) are constants. These equations can have one, two, or no real solutions.
The quadratic formula is a powerful tool to find these solutions. It is given by \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].
This formula might look complicated at first, but breaking it down step-by-step makes it easier.

• Identify the coefficients \(a\), \(b\), and \(c\).


• Compute the discriminant \(b^2 - 4ac\).


• Substitute the values into the formula.


• Simplify to get the solutions.

In the given equation, \[2x^2 + 3x - 1 = 0\], we identified \(a = 2\), \(b = 3\), and \(c = -1\).
Next, we calculated the discriminant, substituted everything into the formula, and found the solutions.

discriminant

The discriminant is a key part of the quadratic formula. It helps in determining the nature and number of the solutions.
The discriminant \(D\) is represented as \(D = b^2 - 4ac\).

• If \(D > 0\), there are two distinct real solutions.


• If \(D = 0\), there is exactly one real solution.


• If \(D < 0\), there are no real solutions, only complex ones.

In the given example, we calculated the discriminant as follows: \[3^2 - 4(2)(-1) = 9 + 8 = 17\].
Since \(17 > 0\), we know there are two distinct real solutions for our equation. This understanding makes it easier to predict the number of solutions before solving the equation fully.

coefficients

Coefficients in a quadratic equation are the numbers in front of the variables \(x^2\), \(x\), and the constant term.
They are crucial for solving the equation using the quadratic formula.

• In the standard form \(ax^2 + bx + c = 0\), \(a\) is the coefficient of \(x^2\), \(b\) is the coefficient of \(x\), and \(c\) is the constant term.


• These coefficients help in setting up the quadratic formula.


• We use them to calculate the discriminant.


• Finally, they are substituted back into the quadratic formula to find the solutions.

In our example, we have \(a = 2\), \(b = 3\), and \(c = -1\).
Identifying these correctly allowed us to proceed with solving the equation accurately using the quadratic formula.