/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 50 \(-x^{2}+6 x+16=0\)... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 50

\(-x^{2}+6 x+16=0\)

Short Answer

Expert verified
The solutions are \(x = -2\) and \(x = 8\).

Step by step solution

01

Identify the coefficients

First, identify the coefficients from the quadratic equation. The equation is \(-x^{2}+6x+16=0\). We rewrite it as \(-1x^{2}+6x+16=0\). The coefficients are \(a = -1\), \(b = 6\), and \(c = 16\).
02

Apply the quadratic formula

The quadratic formula is \(x=\frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\). Substitute the values \(a = -1\), \(b = 6\), and \(c = 16\) into the formula.
03

Calculate the discriminant

Calculate the discriminant using \(b^{2}-4ac\): \[b^{2} - 4ac = 6^{2} - 4(-1)(16) = 36 + 64 = 100\]
04

Solve for x using the quadratic formula

Substitute the values into the quadratic formula to solve for \(x\):\[x = \frac{-6 \pm \sqrt{100}}{-2}\]. Simplifying, we find \(\sqrt{100} = 10\), so the equation becomes \[x = \frac{-6 \pm 10}{-2}\].
05

Calculate the solutions

We have two solutions: 1) \(x = \frac{-6 + 10}{-2} = \frac{4}{-2} = -2\)2) \(x = \frac{-6 - 10}{-2} = \frac{-16}{-2} = 8\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Formula
The quadratic formula is a universal method used to find the solutions, or roots, of a quadratic equation. A quadratic equation is generally expressed in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants with \(a eq 0\). The quadratic formula is:
  • \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
This formula provides the solutions for \(x\) by using the coefficients from the equation. The results include possible real or complex numbers depending on the value within the square root, known as the discriminant. Adopting the quadratic formula allows for a standardized approach to solve any quadratic equation efficiently.
Discriminant
The discriminant in a quadratic equation helps in determining the nature of the roots. It is the expression under the square root in the quadratic formula: \(b^2 - 4ac\). The value of the discriminant points to the type of solutions:
  • If \(b^2 - 4ac > 0\), there are two distinct real solutions.
  • If \(b^2 - 4ac = 0\), there is exactly one real solution, often referred to as a repeated root.
  • If \(b^2 - 4ac < 0\), the equation has two complex solutions.
Evaluating the discriminant first allows you to anticipate whether the solutions will be real or complex, which is a helpful step when solving quadratic equations using the quadratic formula.
Solving Quadratic Equations
To solve quadratic equations like \(-x^2 + 6x + 16 = 0\), one commonly uses the quadratic formula. Here鈥檚 a step-by-step approach:
  • Identify the coefficients \(a\), \(b\), and \(c\). In this equation: \(a = -1\), \(b = 6\), \(c = 16\).
  • Calculate the discriminant \(b^2 - 4ac\). For this example, it is \(6^2 - 4(-1)(16) = 36 + 64 = 100\).
  • Check the discriminant to determine the nature of the roots. Since \(100 > 0\), expect two different real solutions.
  • Substitute back into the quadratic formula: \(x = \frac{-6 \pm \sqrt{100}}{-2}\).
  • Simplify the equation by evaluating the square root and performing arithmetic operations: \(x = \frac{-6 \pm 10}{-2}\). This gives the roots: \(x = -2\) and \(x = 8\).
This method is systematic and guarantees finding the correct solutions by following through the structured formula.

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Most popular questions from this chapter

\(x^{2}+8 x=20\)

One number is 2 larger than another number. The sum of their squares is 100 . Find the numbers.

$$ \text { For Problems } 41-70 \text {, solve each equation. (Objective } 3 \text { ) } $$ $$ x^{3}+10 x^{2}+25 x=0 $$

$$ \text { For Problems } 41-70 \text {, solve each equation. (Objective } 3 \text { ) } $$ $$ (2 n-3)(7 n+1)=0 $$

The length of one leg of a right triangle is 3 centimeters more than the length of the other leg. The length of the hypotenuse is 15 centimeters. Find the lengths of the two legs.
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