Question

Solve the quadratic equation using any method. Find only real solutions. $$-3 x^{2}-2=7 x$$

   Solve the quadratic equation using any method. Find only real solutions.
$$-3 x^{2}-2=7 x$$

Precalculus : Building Concepts and Connections

Precalculus : Building Concepts and Connections

Revathi Narasimhan 1st Edition

Chapter 2, Problem 60

Instant Answer

Solved by Expert Jacey Greenwood

05/08/2020

Step 1

The standard form of a quadratic equation is $ax^2 + bx + c = 0$. So, we move $7x$ to the left side of the equation: $$-3x^2 - 7x - 2 = 0$$ Show more…

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Solve the quadratic equation using any method. Find only real solutions. $$-3 x^{2}-2=7 x$$

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Key Concepts

Real Number Solutions

When solving quadratic equations, the focus on real number solutions involves ensuring that the discriminant (b² - 4ac) is non-negative, because a negative discriminant would result in complex roots. This condition is essential when the problem specifies that only real solutions should be considered.

Quadratic Formula

The quadratic formula, x = (-b ± ?(b² - 4ac)) / (2a), is a comprehensive method for solving any quadratic equation. It directly uses the coefficients from the standard form to determine the solutions, making it a widely applicable technique.

Discriminant

The discriminant, given by b² - 4ac, is a key component of the quadratic formula that determines the nature of the roots of the quadratic equation. A non-negative discriminant indicates the existence of real roots, with a discriminant of zero yielding a repeated real root and a positive discriminant providing two distinct real roots.

Quadratic Equations

A quadratic equation is a polynomial equation of degree two, typically represented in the standard form ax² + bx + c = 0, where a, b, and c are constants and a is not equal to zero. This form allows for various methods of solving, including factoring, completing the square, and using the quadratic formula.

Standard Form

Converting a given quadratic equation into the standard form is crucial because it standardizes the equation for application of systematic solving methods. This involves rearranging all terms so that one side of the equation is zero, making it easier to identify the coefficients a, b, and c.

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