Problem 104 - Algebra for College Students

Question

Determine the number of real solutions to each equation by examining the calculator graph of \(y=a x^{2}+b x+c .\) Use the discriminant to check your conclusions. $$-2 x^{2}-403=0$$

Step-by-Step Solution

Answer

The quadratic equation has no real solutions.

Step 1: Rewrite the Equation in Standard Form

The given equation is \[-2x^2 - 403 = 0 \]. Rewrite it in the form \[ ax^2 + bx + c = 0 \] to identify coefficients. In this case, \[ a = -2 \], \[ b = 0 \], and \[ c = -403 \].

Step 2: Identify the Coefficients

From the standard form equation, note that \[ a = -2 \], \[ b = 0 \], and \[ c = -403 \].

Step 3: Calculate the Discriminant

The discriminant of a quadratic equation \[ ax^2 + bx + c = 0 \] is given by \[ \Delta = b^2 - 4ac \]. Substitute the values of \[ a, b, \text{ and } c:\] \[ \Delta = 0^2 - 4(-2)(-403) \]. Therefore, \[ \Delta = -8 \times 403 = -3224 \].

Step 4: Interpret the Discriminant

Since the discriminant \[ \Delta = -3224 \] is less than zero, this means the quadratic equation has no real solutions.

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation that involve the square of the variable. The general form is \[ax^2 + bx + c = 0,\] where - \(a\), \(b\), and \(c\) are constants - \(a eq 0\) (otherwise, the equation is linear, not quadratic).The variable here is \(x\), and the highest power of \(x\) is 2. Some unique characteristics differentiate quadratic equations from linear equations. The most noticeable feature is the parabola shape of their graph when plotted on a coordinate system.

Real Solutions

Solutions to quadratic equations are the values of \(x\) that make the equation true when substituted back in. These are also known as the roots of the equation. Solutions can be real numbers, complex numbers, or sometimes no solutions at all. Whether a quadratic equation has real solutions depends on the value of the discriminant \(∆\). The discriminant is found using the formula \[Δ = b^2 - 4ac\]. Depending on the value of \(Δ\):

• If \(Δ > 0\), there are two distinct real solutions.
• If \(Δ = 0\), there is exactly one real solution.
• If \(Δ < 0\), there are no real solutions (only complex solutions).

Standard Form of Quadratic Equations

The standard form of a quadratic equation is an equation written as \[ax^2 + bx + c = 0,\]where \(a\), \(b\), and \(c\) represent known values (coefficients) and \(x\) is the variable. It's essential to write the quadratic equation in this form to easily identify the coefficients for use in various methods of solving the equation. Whether you're factoring, using the quadratic formula, or graphing, having it in standard form simplifies these processes.

Identifying Coefficients

Coefficients in a quadratic equation are the constants \(a\), \(b\), and \(c\) in the standard form \[ax^2 + bx + c = 0\]. Identifying these coefficients is crucial for solving quadratic equations, whether you want to factor the equation, complete the square, or use the quadratic formula. Let's take the example equation \[-2x^2 - 403 = 0\]First, rewrite it in standard form: \[-2x^2 + 0x - 403 = 0\]. Here,

• a = -2
• b = 0
• c = -403
.

Recognizing these coefficients allows us to substitute them into the discriminant formula and determine the nature of the solutions.