Question

Which of the following could be a value of \(x\) in this equation? $$ 8 x^{2}=-16 x-2 $$ I. \(-1-\frac{\sqrt{3}}{2}\) II. \(\frac{1}{2}(-2-\sqrt{6})\) III. \(\frac{1}{2}(\sqrt{3}-2)\) (A) I only (B) II only (C) \(\mid\) and IIl only (D) \(\quad\) II and III only

Short answer

(C) I and III only

Step-by-step solution

Rearrange the Equation

Start by rearranging the given quadratic equation into standard form. The original equation is \(8x^2 = -16x - 2\). Move all terms to one side to get \(8x^2 + 16x + 2 = 0\).

Quadratic Formula Application

The quadratic formula can be used to find the roots of a quadratic equation in the form \(ax^2 + bx + c = 0\). Here, \(a = 8\), \(b = 16\), and \(c = 2\). The formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Calculate the Discriminant

Calculate the discriminant \(\Delta\) as \(b^2 - 4ac = 16^2 - 4 \cdot 8 \cdot 2 = 256 - 64 = 192\).

Find the Roots

Use the quadratic formula with the computed discriminant. The roots are \(x = \frac{-16 \pm \sqrt{192}}{16}\). Simplify to \(x = \frac{-16 \pm 8\sqrt{3}}{16}\), which reduces to \(x = \frac{-2 \pm \sqrt{3}}{2}\). The two potential solutions are \(-1 - \frac{\sqrt{3}}{2}\) and \(-1 + \frac{\sqrt{3}}{2}\).

Verify the Choices

Compare the solutions against the given options. Option I is \(-1 - \frac{\sqrt{3}}{2}\), matching one root. Option II is another transformation of the same root, \(-1 - \frac{\sqrt{3}}{2}\). Option III is \(\frac{1}{2}(\sqrt{3}-2)\), equivalent to \(-1 + \frac{\sqrt{3}}{2}\). Both represent the roots found.

Key concepts

Quadratic equations

Quadratic equations are key algebraic expressions that students encounter in math. These equations are primarily in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). The term \(ax^2\) is known as the quadratic term, \(bx\) is the linear term, and \(c\) is the constant term. This specific form is called the standard form of a quadratic equation.

To solve quadratic equations, students often need to find the values of \(x\) that make the equation true. These solutions are known as the roots of the equation. Quadratic equations can have two solutions, one solution, or no real solution because they are second-degree polynomials.

There are multiple methods to solve quadratic equations, including:

• Factoring the quadratic expression.
• Using the quadratic formula.
• Completing the square.
• Graphing the equation to find where it crosses the \(x\)-axis.

Understanding these equations is crucial, as they frequently appear in various problems on the PSAT Math section, testing students' knowledge and flexibility in applying algebraic methods.

Quadratic formula

The quadratic formula is a universal tool to solve any quadratic equation, especially when other methods like factoring are not feasible. The formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \(a\), \(b\), and \(c\) are coefficients from the quadratic equation \(ax^2 + bx + c = 0\). This formula calculates the roots (solutions for \(x\)) by considering these coefficients.

The symbol \(\pm\) indicates that the formula will yield two solutions: one involving addition and the other subtraction of the square root term. These two potential solutions correspond to the parabola's intersection points with the \(x\)-axis.

Using the quadratic formula follows a straightforward process:

• Identify values of \(a\), \(b\), and \(c\) from the equation.
• Calculate the discriminant, \(b^2 - 4ac\).
• Substitute \(a\), \(b\), and \(c\) into the quadratic formula and solve for \(x\).

The quadratic formula is essential, especially on standardized tests like the PSAT, as it provides a reliable method to solve any quadratic equation without trial and error.

Discriminant calculation

The discriminant is a part of the quadratic formula and plays a vital role in determining the nature of the roots of a quadratic equation. It is represented as \(b^2 - 4ac\). This value can tell us about the type and number of solutions the quadratic equation will have.

Here's how the discriminant affects the roots:

• If the discriminant is positive, \(b^2 - 4ac > 0\), the quadratic equation has two distinct real roots.
• If the discriminant is zero, \(b^2 - 4ac = 0\), there is exactly one real root, meaning the parabola touches the \(x\)-axis at a single point, indicating a repeated or double root.
• If the discriminant is negative, \(b^2 - 4ac < 0\), the equation has no real roots, as the parabola does not intersect the \(x\)-axis. Instead, it has two complex roots.

Understanding how to compute and interpret the discriminant is essential for solving quadratic equations accurately. It not only helps in finding solutions but also provides insights into the equation's broader geometrical implications.