Problem 83 - Algebra 1: Concepts and Skills

Question

Which quadratic equation has the solutions \(x=\frac{-9 \pm \sqrt{81-56}}{4} ?\) $$A.)2 x^{2}+9 x-7=0$$ $$B.)2 x^{2}-9 x+7=0$$ $$C.)2 x^{2}-9 x+7=0$$ $$D.)2 x^{2}+9 x+7=0$$

Step-by-Step Solution

Answer

The quadratic equation which has the solutions \(x=\frac{-9 \pm \sqrt{81-56}}{4}\) is \(2x^2+9x-7=0\), thus the correct option is A.)

Step 1: Identify the values of a, b and c from the given solution

The solution \(x=\frac{-9 \pm \sqrt{81-56}}{4}\) is in the form of quadratic formula \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\). Comparing the two, it can be deduced that a=2, b=9 and c=-7. Here a is the coefficient of \(x^2\), b is the coefficient of x and c is the constant term.

Step 2: Form the quadratic equation from identified values

Using the obtained values of a, b and c, the quadratic equation becomes \(2x^2+9x-7=0\) .

Step 3: Match the formed equation with the options

Looking into the options given, equation A.) \(2x^2+9x-7=0\) matches with the equation formed. Thus, A.) is the correct option.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of any quadratic equation. A quadratic equation generally has the form \(ax^2 + bx + c = 0\). The solutions to this equation are given by the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here's how it works:

• The variable \(a\) represents the coefficient of \(x^2\).
• The variable \(b\) is the coefficient of \(x\).
• The variable \(c\) is the constant term.

By substituting these values into the quadratic formula, you can find the solutions \(x\) that satisfy the equation. Remember, the "±" means you'll generally find two solutions—one using addition and one using subtraction. This formula is extremely useful as it works for all kinds of quadratic equations, regardless of whether the solutions are real or complex.

Solutions of Quadratic Equations

Finding solutions for quadratic equations involves determining the values of \(x\) that make the equation \(ax^2 + bx + c = 0\) true. The solutions can be discovered through several methods, but the quadratic formula is one of the most reliable. Here’s what happens when we solve a quadratic equation:

• If the term \(b^2 - 4ac\), known as the discriminant, is positive, you get two real and distinct solutions.
• When the discriminant is zero, you get exactly one real solution, which is sometimes referred to as a repeated root.
• If the discriminant is negative, the solutions are complex (involving imaginary numbers).

In many school problems, solutions are real and simplified logically. However, understanding that solutions can also be complex is key to mastering more advanced mathematical concepts. It's important to plug the solutions back into the original equation to verify their correctness, ensuring all calculations have been accurately performed.

Coefficients in Quadratic Equations

Coefficients play a crucial role in forming and solving quadratic equations. Each term in the quadratic equation \(ax^2 + bx + c = 0\) has a significant coefficient:

• \(a\) is the coefficient of \(x^2\), which determines the parabola's "width" and direction (opens upwards if \(a > 0\), and downwards if \(a < 0\)).
• \(b\) is the coefficient of \(x\), influencing the slope or the line's tilt as it crosses the x-axis.
• \(c\) is the constant term, affecting the parabola's vertical position on the graph.

Identifying these coefficients from the given quadratic formula solution allows for the construction of the specific quadratic equation. In our example, the values \(a = 2\), \(b = 9\), and \(c = -7\) help form the quadratic equation that can be further verified against given options. Understanding these elements not only aids in building equations but also in graphically interpreting the nature and solution set of the quadratic equation.