Question

question 3 (mandatory) (1 point) saved
use the discriminant to determine the number of roots of $-4(x + 5)^2 - 1 = 0$.
\bigcirc a) two
\bigcirc b) three
\bigcirc c) zero
\bigcirc d) one

Explanation:

Step1: Rewrite the equation

First, rewrite the given equation \(-4(x + 5)^2 - 1 = 0\) in standard quadratic form \(ax^2+bx + c = 0\). Expand \((x + 5)^2=x^2 + 10x+25\), so the equation becomes \(-4(x^2 + 10x + 25)-1=0\), which simplifies to \(-4x^2-40x - 100 - 1 = 0\), or \(-4x^2-40x - 101 = 0\). Here, \(a=-4\), \(b = - 40\), \(c=-101\).

Step2: Calculate the discriminant

The discriminant of a quadratic equation \(ax^2+bx + c = 0\) is given by \(D=b^2-4ac\). Substitute \(a=-4\), \(b=-40\), \(c = - 101\) into the formula: \(D=(-40)^2-4\times(-4)\times(-101)\). Calculate \((-40)^2 = 1600\), and \(4\times(-4)\times(-101)=4\times404 = 1616\). Then \(D=1600 - 1616=-16\).

Step3: Analyze the discriminant

Since the discriminant \(D=-16<0\), a quadratic equation with a negative discriminant has no real roots (zero real roots).

Answer:

c) zero